Theorem 19.35-1 1460

Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.35-1	⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))

Proof of Theorem 19.35-1

Step	Hyp	Ref	Expression
1		19.29 1456	. . 3 ⊢ ((∀xφ ∧ ∃x(φ → ψ)) → ∃x(φ ∧ (φ → ψ)))
2		pm3.35 326	. . . 4 ⊢ ((φ ∧ (φ → ψ)) → ψ)
3	2	eximi 1437	. . 3 ⊢ (∃x(φ ∧ (φ → ψ)) → ∃xψ)
4	1, 3	syl 13	. 2 ⊢ ((∀xφ ∧ ∃x(φ → ψ)) → ∃xψ)
5	4	expcom 107	1 ⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))

Syntax hints:   → wi 4   ∧ wa 95  ∀wal 1281  ∃wex 1328

This theorem is referenced by:  19.35i  1461  19.25  1462  19.36-1  1504  19.37-1  1505  spimt  1563  sbequi  1659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-5 1282  ax-gen 1284  ax-ie1 1329  ax-ie2 1330  ax-4 1349  ax-ial 1376

This theorem depends on definitions:  df-bi 108