Theorem 19.36i 2086

Description: Inference associated with 19.36 2085. See 19.36iv 1892 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 2085	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	mpbi 219	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  spimv1  2101  spim  2242  vtoclf  3231  bj-vtoclf  32100