Theorem eexinst11 37754

Description: exinst11 37872 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eexinst11.1	⊢ (𝜑 → ∃𝑥𝜓)
eexinst11.2	⊢ (𝜑 → (𝜓 → 𝜒))
eexinst11.3	⊢ (𝜑 → ∀𝑥𝜑)
eexinst11.4	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
eexinst11	⊢ (𝜑 → 𝜒)

Proof of Theorem eexinst11

Step	Hyp	Ref	Expression
1		eexinst11.1	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
2		eexinst11.3	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3		eexinst11.4	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
4		eexinst11.2	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
5	2, 3, 4	exlimdh 2134	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
6	1, 5	syl5com 31	. 2 ⊢ (𝜑 → (𝜑 → 𝜒))
7	6	pm2.43i 50	1 ⊢ (𝜑 → 𝜒)

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:  vk15.4j  37755  exinst11  37872