Theorem bj-exlimmpi 37580

Description: Lemma for bj-vtoclg1f1 37585 (an instance of this lemma is a version of bj-vtoclg1f1 37585 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 21	. 2 ⊢ (𝜒 → 𝜓)
5	1, 4	exlimi 2256	1 ⊢ (∃𝑥𝜒 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  bj-vtoclg1f1  37585  bj-vtoclg1f  37586  bj-vtoclg1fv  37587