Theorem exlimd 2209

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 2207	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2196	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 250	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-ex 1780  df-nf 1784

This theorem is referenced by:  exlimimdd  2210  exlimdh  2284  equs5  2457  moexexlem  2620  2eu6  2650  ceqsalgALT  3507  alxfr  5406  copsex2t  5493  mosubopt  5511  ov3  7574  tz7.48-1  8447  ac6c4  10480  fsum2dlem  15722  fprod2dlem  15930  gsum2d2lem  19884  exlimim  36528  exellim  36530  wl-lem-moexsb  36738  exlimddvf  37294  mnringmulrcld  43291  fourierdlem31  45154  or2expropbi  46044  ich2exprop  46439  ichreuopeq  46441  reuopreuprim  46494