Theorem bj-exlime 32609

Description: Variant of exlimih 2148 where the non-freeness of 𝑥 in 𝜓 is expressed using an existential quantifier, thus requiring fewer axioms. (Contributed by BJ, 17-Mar-2020.)

Hypotheses

Ref	Expression
bj-exlime.1	⊢ (∃𝑥𝜓 → 𝜓)
bj-exlime.2	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
bj-exlime	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem bj-exlime

Step	Hyp	Ref	Expression
1		bj-exlime.2	. . 3 ⊢ (𝜑 → 𝜓)
2	1	eximi 1762	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
3		bj-exlime.1	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl 17	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-cbvexiw  32659