Theorem 19.36-1 1725

Description: Closed form of 19.36i 1724. 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 1677	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.36-1.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	2	19.9 1697	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	1, 3	imbitrdi 161	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1400  Ⅎwnf 1513  ∃wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  vtocl2  2878  vtocl3  2879  spcimgft  2901