Theorem 19.37iv 1982

Description: Inference associated with 19.37v 2031. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 2001. (Revised by Rohan Ridenour, 15-Apr-2022.)

Hypothesis

Ref	Expression
19.37iv.1	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.37iv	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37iv

Step	Hyp	Ref	Expression
1		19.37iv.1	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		19.37imv 1981	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  axprg  5399  bnd  9968  zfcndinf  10716  bnj1093  35666  bnj1186  35693  relopabVD  46025  elpglem2  51022