Theorem exlimdh 2327

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 2184	. 2 ⊢ Ⅎ𝑥𝜑
3		exlimdh.2	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
4	3	nf5i 2184	. 2 ⊢ Ⅎ𝑥𝜒
5		exlimdh.3	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 4, 5	exlimd 2257	1 ⊢ (𝜑 → (∃𝑥𝜓 → 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:  exlimexi  45293  eexinst01  45295  eexinst11  45296