Theorem exlimd 2256

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 2254	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2243	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 254	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1802  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-ex 1803  df-nf 1807

This theorem is referenced by:  exlimimdd  2257  exlimdh  2327  equs5  2494  moexexlem  2656  2eu6  2686  ceqsalgALT  3493  alxfr  5369  copsex2t  5466  mosubopt  5484  ov3  7563  tz7.48-1  8418  ac6c4  10453  fsum2dlem  15811  fprod2dlem  16024  gsum2d2lem  20034  exlimim  37848  exellim  37850  wl-lem-moexsb  38083  exlimddvf  38632  mnringmulrcld  44816  fourierdlem31  46710  or2expropbi  47626  ich2exprop  48075  ichreuopeq  48077  reuopreuprim  48130