Theorem exlimd 2210

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 2208	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2197	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 250	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-ex 1781  df-nf 1785

This theorem is referenced by:  exlimimdd  2211  exlimdh  2285  equs5  2458  moexexlem  2621  2eu6  2651  ceqsalgALT  3508  alxfr  5405  copsex2t  5492  mosubopt  5510  ov3  7574  tz7.48-1  8449  ac6c4  10482  fsum2dlem  15723  fprod2dlem  15931  gsum2d2lem  19889  exlimim  36689  exellim  36691  wl-lem-moexsb  36899  exlimddvf  37455  mnringmulrcld  43452  fourierdlem31  45315  or2expropbi  46205  ich2exprop  46600  ichreuopeq  46602  reuopreuprim  46655