Theorem bj-exlimvmpi 37527

Description: A Fol lemma (exlimiv 1960 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 1960	1 ⊢ (∃𝑥𝜒 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940

This theorem depends on definitions:  df-bi 210  df-ex 1810

This theorem is referenced by:  bj-vtoclg  37536