Theorem exlimd 2257

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 2255	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2244	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 254	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  exlimimdd  2258  exlimdh  2328  equs5  2495  moexexlem  2657  2eu6  2687  ceqsalgALT  3494  alxfr  5383  copsex2t  5480  mosubopt  5498  ov3  7586  tz7.48-1  8439  ac6c4  10483  fsum2dlem  15847  fprod2dlem  16060  gsum2d2lem  20074  exlimim  38029  exellim  38031  wl-lem-moexsb  38264  exlimddvf  38811  mnringmulrcld  44993  fourierdlem31  46893  or2expropbi  47812  ich2exprop  48261  ichreuopeq  48263  reuopreuprim  48316