Theorem 19.37aiv 1686

Description: Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.37aiv.1	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.37aiv	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37aiv

Step	Hyp	Ref	Expression
1		19.37aiv.1	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		nfv 1539	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	19.37-1 1685	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  eqvinc  2883  limom  4646