Theorem exlimexi 41230

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 2144	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2		exlimexi.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3		exlimexi.2	. . 3 ⊢ (∃𝑥𝜑 → (𝜑 → 𝜓))
4	1, 2, 3	exlimdh 2294	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥𝜑 → 𝜓))
5	4	pm2.43i 52	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786

This theorem is referenced by:  sb5ALT  41231  exinst  41330