Theorem eexinst01 44970

Description: exinst01 45069 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eexinst01.1	⊢ ∃𝑥𝜓
eexinst01.2	⊢ (𝜑 → (𝜓 → 𝜒))
eexinst01.3	⊢ (𝜑 → ∀𝑥𝜑)
eexinst01.4	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
eexinst01	⊢ (𝜑 → 𝜒)

Proof of Theorem eexinst01

Step	Hyp	Ref	Expression
1		eexinst01.1	. 2 ⊢ ∃𝑥𝜓
2		eexinst01.3	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		eexinst01.4	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
4		eexinst01.2	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	2, 3, 4	exlimdh 2301	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
6	1, 5	mpi 20	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  exinst01  45069