Theorem 19.36i 2243

Description: Inference associated with 19.36 2242. See 19.36iv 1953 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)

Hypotheses

Ref	Expression
19.36.1	⊢ Ⅎ𝑥𝜓
19.36i.2	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.36i	⊢ (∀𝑥𝜑 → 𝜓)

Proof of Theorem 19.36i

Step	Hyp	Ref	Expression
1		19.36i.2	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		19.36.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	2	19.36 2242	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	mpbi 231	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  spimfv  2251  spim  2395  bj-vtoclf  37268