Theorem 19.36i 2232

Description: Inference associated with 19.36 2231. See 19.36iv 1946 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 2231	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	mpbi 230	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by:  spimfv  2240  spim  2395  vtoclfOLD  3577  bj-vtoclf  36881