Theorem 19.35-1 1512

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 1508	. . 3 ⊢ ((∀xφ ∧ ∃x(φ → ψ)) → ∃x(φ ∧ (φ → ψ)))
2		pm3.35 329	. . . 4 ⊢ ((φ ∧ (φ → ψ)) → ψ)
3	2	eximi 1488	. . 3 ⊢ (∃x(φ ∧ (φ → ψ)) → ∃xψ)
4	1, 3	syl 14	. 2 ⊢ ((∀xφ ∧ ∃x(φ → ψ)) → ∃xψ)
5	4	expcom 109	1 ⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.35i  1513  19.25  1514  19.36-1  1560  19.37-1  1561  spimt  1621  sbequi  1717