Theorem exlimdh 2290

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 2144	. 2 ⊢ Ⅎ𝑥𝜑
3		exlimdh.2	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
4	3	nf5i 2144	. 2 ⊢ Ⅎ𝑥𝜒
5		exlimdh.3	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 4, 5	exlimd 2214	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by:  exlimexi  42033  eexinst01  42035  eexinst11  42036