Theorem eexinst11 44971

Description: exinst11 45070 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 2301	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
6	1, 5	syl5com 31	. 2 ⊢ (𝜑 → (𝜑 → 𝜒))
7	6	pm2.43i 52	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  vk15.4j  44972  exinst11  45070