Theorem 19.37-1 1726

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 1607	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3		19.35-1 1677	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	biimtrrid 153	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

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:  19.37aiv  1727  spcimegft  2903  eqvincg  2950  reldmm  4995