Theorem bj-exlimvmpi 37579

Description: A Fol lemma (exlimiv 1963 followed by mpi 21). (Contributed by BJ, 2-Jul-2022.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-exlimvmpi.maj	⊢ (𝜒 → (𝜑 → 𝜓))
bj-exlimvmpi.min	⊢ 𝜑

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem bj-exlimvmpi

Step	Hyp	Ref	Expression
1		bj-exlimvmpi.min	. . 3 ⊢ 𝜑
2		bj-exlimvmpi.maj	. . 3 ⊢ (𝜒 → (𝜑 → 𝜓))
3	1, 2	mpi 21	. 2 ⊢ (𝜒 → 𝜓)
4	3	exlimiv 1963	1 ⊢ (∃𝑥𝜒 → 𝜓)

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

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

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

This theorem is used by:  bj-vtoclg  37588