Theorem 19.37-1 1564

Description: One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)

Hypothesis

Ref	Expression
19.37-1.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.37-1	⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.37-1

Step	Hyp	Ref	Expression
1		19.37-1.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.3 1446	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3		19.35-1 1515	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	syl5bir 142	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  19.37aiv  1565  spcimegft  2631  eqvincg  2668