Theorem bj-exlimmpbir 34225

Description: Lemma for theorems of the vtoclg 3568 family. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimmpbir.nf	⊢ Ⅎ𝑥𝜑
bj-exlimmpbir.maj	⊢ (𝜒 → (𝜑 ↔ 𝜓))
bj-exlimmpbir.min	⊢ 𝜓

Assertion

Ref	Expression
bj-exlimmpbir	⊢ (∃𝑥𝜒 → 𝜑)

Proof of Theorem bj-exlimmpbir

Step	Hyp	Ref	Expression
1		bj-exlimmpbir.nf	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-exlimmpbir.min	. . 3 ⊢ 𝜓
3		bj-exlimmpbir.maj	. . 3 ⊢ (𝜒 → (𝜑 ↔ 𝜓))
4	2, 3	mpbiri 260	. 2 ⊢ (𝜒 → 𝜑)
5	1, 4	exlimi 2212	1 ⊢ (∃𝑥𝜒 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781

This theorem is referenced by: (None)