Theorem eexinst01 45295

Description: exinst01 45394 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 2327	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
6	1, 5	mpi 21	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  vk15.4j  45297  exinst01  45394