Theorem bj-sylge 37428

Description: Dual statement of sylg 1856 (the final "e" in the label stands for "existential (version of sylg 1856)". Variant of exlimih 2322. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

Ref	Expression
bj-sylge.nf	⊢ (∃𝑥𝜑 → 𝜓)
bj-sylge.maj	⊢ (𝜒 → 𝜑)

Assertion

Ref	Expression
bj-sylge	⊢ (∃𝑥𝜒 → 𝜓)

Proof of Theorem bj-sylge

Step	Hyp	Ref	Expression
1		bj-sylge.maj	. . 3 ⊢ (𝜒 → 𝜑)
2	1	eximi 1868	. 2 ⊢ (∃𝑥𝜒 → ∃𝑥𝜑)
3		bj-sylge.nf	. 2 ⊢ (∃𝑥𝜑 → 𝜓)
4	2, 3	syl 18	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

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

This theorem is used by:  bj-cbvexiw  37493  bj-hbex  37536