Theorem 19.36-1 1560

Description: Closed form of 19.36i 1559. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)

Hypothesis

Ref	Expression
19.36-1.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.36-1	⊢ (∃x(φ → ψ) → (∀xφ → ψ))

Proof of Theorem 19.36-1

Step	Hyp	Ref	Expression
1		19.35-1 1512	. 2 ⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))
2		19.36-1.1	. . 3 ⊢ Ⅎxψ
3	2	19.9 1532	. 2 ⊢ (∃xψ ↔ ψ)
4	1, 3	syl6ib 150	1 ⊢ (∃x(φ → ψ) → (∀xφ → ψ))

Syntax hints:   → wi 4  ∀wal 1240  Ⅎwnf 1346  ∃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  df-nf 1347

This theorem is referenced by:  vtocl2  2603  vtocl3  2604  spcimgft  2623