Theorem 19.37aiv 1663

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 1516	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	19.37-1 1662	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
4	1, 3	ax-mp 5	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  eqvinc  2849  limom  4591