Theorem exlimd 2254

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 2252	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2241	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 254	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1809  Ⅎwnf 1813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810  df-nf 1814

This theorem is referenced by:  exlimimdd  2255  exlimdh  2325  equs5  2492  moexexlem  2654  2eu6  2684  ceqsalgALT  3491  alxfr  5380  copsex2t  5477  mosubopt  5495  ov3  7575  tz7.48-1  8431  ac6c4  10466  fsum2dlem  15823  fprod2dlem  16036  gsum2d2lem  20044  exlimim  37966  exellim  37968  wl-lem-moexsb  38201  exlimddvf  38748  mnringmulrcld  44932  fourierdlem31  46832  or2expropbi  47748  ich2exprop  48197  ichreuopeq  48199  reuopreuprim  48252