Theorem exlimexi 37751

Description: Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
exlimexi.1	⊢ (𝜓 → ∀𝑥𝜓)
exlimexi.2	⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))

Assertion

Ref	Expression
exlimexi	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem exlimexi

Step	Hyp	Ref	Expression
1		hbe1 2008	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		exlimexi.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		exlimexi.2	. . 3 ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))
4	1, 2, 3	exlimdh 2134	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥𝜑 → 𝜓))
5	4	pm2.43i 50	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  sb5ALT  37752  exinst  37870