Theorem exlimd 2125

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 2123	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2110	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	syl6ib 241	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1744  Ⅎwnf 1748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750

This theorem is referenced by:  exlimdd  2126  exlimdh  2187  equs5  2379  moexex  2570  2eu6  2587  exists2  2591  ceqsalgALT  3262  alxfr  4908  copsex2t  4986  mosubopt  5001  ovmpt2df  6834  ov3  6839  tz7.48-1  7583  ac6c4  9341  fsum2dlem  14545  fprod2dlem  14754  gsum2d2lem  18418  padct  29625  exlimim  33319  exellim  33322  wl-lem-moexsb  33480  exlimddvf  34056  stoweidlem27  40562  fourierdlem31  40673  intsaluni  40865  isomenndlem  41065