Theorem 19.37iv 1953

Description: Inference associated with 19.37v 1996. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-6 1972. (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 1952	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  bnd  9581  zfcndinf  10305  bnj1093  32860  bnj1186  32887  relopabVD  42410  elpglem2  46303