Theorem 19.36iv 1949

Description: Inference associated with 19.36v 1990. Version of 19.36i 2223 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.) Remove dependency on ax-6 1970. (Revised by Rohan Ridenour, 15-Apr-2022.)

Hypothesis

Ref	Expression
19.36iv.1	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.36iv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36iv

Step	Hyp	Ref	Expression
1		19.36iv.1	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		19.36imv 1947	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912

This theorem depends on definitions:  df-bi 206  df-ex 1781

This theorem is referenced by:  spimvALT  2389  zfcndext  10614