Theorem exlimd 2214

Description: Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)

Hypotheses

Ref	Expression
exlimd.1	⊢ Ⅎ𝑥𝜑
exlimd.2	⊢ Ⅎ𝑥𝜒
exlimd.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

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

Proof of Theorem exlimd

Step	Hyp	Ref	Expression
1		exlimd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		exlimd.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	eximd 2212	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2201	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	syl6ib 253	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1776  Ⅎwnf 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-ex 1777  df-nf 1781

This theorem is referenced by:  exlimimdd  2215  exlimddOLD  2217  exlimdh  2294  equs5  2479  moexexlem  2707  2eu6  2740  ceqsalgALT  3530  alxfr  5299  copsex2t  5375  mosubopt  5392  ov3  7305  tz7.48-1  8073  ac6c4  9897  fsum2dlem  15119  fprod2dlem  15328  gsum2d2lem  19087  exlimim  34617  exellim  34619  wl-lem-moexsb  34798  exlimddvf  35393  fourierdlem31  42417  or2expropbi  43263  ich2exprop  43627  ichreuopeq  43629  reuopreuprim  43682