Theorem exlimd 2216

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 2214	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2203	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 251	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 1908  ax-6 1965  ax-7 2005  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781

This theorem is referenced by:  exlimimdd  2217  exlimdh  2289  equs5  2463  moexexlem  2624  2eu6  2655  ceqsalgALT  3516  alxfr  5413  copsex2t  5503  mosubopt  5520  ov3  7596  tz7.48-1  8482  ac6c4  10519  fsum2dlem  15803  fprod2dlem  16013  gsum2d2lem  20006  exlimim  37325  exellim  37327  wl-lem-moexsb  37549  exlimddvf  38108  mnringmulrcld  44224  fourierdlem31  46094  or2expropbi  46984  ich2exprop  47396  ichreuopeq  47398  reuopreuprim  47451