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It is from discrete mathematics notes. These are part of the biconditional tautologies.

E29. (∀x)A(x) →B ⇔ (∃x)(A(x)→B)

E30. (∃x)A(x) →B ⇔ (∀x)(A(x)→B)

I don't understand how these formulas are said to be equivalent. What is the difference in interpretation between (∀x)A(x) →B and (∀x)(A(x)→B) ? How should I interpret those formulas? and how can I prove it? Thank you!

Mauro ALLEGRANZA
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  • Reagrding you question about the difference in interpretation, you might want to take a look at this question: https://math.stackexchange.com/questions/3146114/logical-equivalence-of-quantified-propositions-involving-if-then – frabala Mar 13 '19 at 15:31
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    Consider a simple example in the language of arithmetic : $\forall x (x=0) \to (1=0)$ is TRUE while $\forall x [(x=0) \to (1=0)]$ is FALSE. – Mauro ALLEGRANZA Mar 13 '19 at 15:34
  • This may be helpful: https://math.stackexchange.com/questions/3051407/is-it-possible-to-convert-forall-quantifiers-to-exists-quantifiers-without/3051463#3051463 – Bram28 Mar 13 '19 at 20:12

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