Theorem bj-exlimmpi 34718

Description: Lemma for bj-vtoclg1f1 34723 (an instance of this lemma is a version of bj-vtoclg1f1 34723 where 𝑥 and 𝑦 are identified). (Contributed by BJ, 30-Apr-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimmpi.nf	⊢ Ⅎ𝑥𝜓
bj-exlimmpi.maj	⊢ (𝜒 → (𝜑 → 𝜓))
bj-exlimmpi.min	⊢ 𝜑

Assertion

Ref	Expression
bj-exlimmpi	⊢ (∃𝑥𝜒 → 𝜓)

Proof of Theorem bj-exlimmpi

Step	Hyp	Ref	Expression
1		bj-exlimmpi.nf	. 2 ⊢ Ⅎ𝑥𝜓
2		bj-exlimmpi.min	. . 3 ⊢ 𝜑
3		bj-exlimmpi.maj	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
4	2, 3	mpi 20	. 2 ⊢ (𝜒 → 𝜓)
5	1, 4	exlimi 2218	1 ⊢ (∃𝑥𝜒 → 𝜓)

Syntax hints:   → wi 4  ∃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 1916  ax-6 1974  ax-7 2019  ax-12 2178

This theorem depends on definitions:  df-bi 210  df-ex 1787  df-nf 1791

This theorem is referenced by:  bj-vtoclg1f1  34723  bj-vtoclg1f  34724  bj-vtoclg1fv  34725