Theorem exlimd 2230

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 2228	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2217	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 252	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  exlimimdd  2231  exlimdh  2301  equs5  2468  moexexlem  2630  2eu6  2661  ceqsalgALT  3469  alxfr  5343  copsex2t  5440  mosubopt  5458  ov3  7526  tz7.48-1  8379  ac6c4  10401  fsum2dlem  15730  fprod2dlem  15943  gsum2d2lem  19946  exlimim  37711  exellim  37713  wl-lem-moexsb  37946  exlimddvf  38495  mnringmulrcld  44679  fourierdlem31  46588  or2expropbi  47504  ich2exprop  47953  ichreuopeq  47955  reuopreuprim  48008