Theorem exlimdh 2323

Description: Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.)

Hypotheses

Ref	Expression
exlimdh.1	⊢ (𝜑 → ∀𝑥𝜑)
exlimdh.2	⊢ (𝜒 → ∀𝑥𝜒)
exlimdh.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimdh	⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Proof of Theorem exlimdh

Step	Hyp	Ref	Expression
1		exlimdh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2182	. 2 ⊢ Ⅎ𝑥𝜑
3		exlimdh.2	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
4	3	nf5i 2182	. 2 ⊢ Ⅎ𝑥𝜒
5		exlimdh.3	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 4, 5	exlimd 2254	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  exlimexi  45643  eexinst01  45645  eexinst11  45646