Theorem exlimd 2071

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 2069	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2057	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	syl6ib 239	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1694  Ⅎwnf 1698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031

This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700

This theorem is referenced by:  exlimdd  2072  exlimdh  2131  equs5  2334  moexex  2524  2eu6  2541  exists2  2545  ceqsalgALT  3199  alxfr  4795  copsex2t  4873  mosubopt  4884  ovmpt2df  6664  ov3  6669  tz7.48-1  7398  ac6c4  9159  fsum2dlem  14285  fprod2dlem  14491  gsum2d2lem  18137  padct  28687  bj-equs5v  31745  exlimim  32164  exellim  32167  wl-lem-moexsb  32328  exlimddvf  32895  stoweidlem27  38720  fourierdlem31  38831  intsaluni  39023  isomenndlem  39220