Theorem exlimd 2085

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 2083	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2070	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	syl6ib 241	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1702  Ⅎwnf 1706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045

This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708

This theorem is referenced by:  exlimdd  2086  exlimdh  2147  equs5  2349  moexex  2539  2eu6  2556  exists2  2560  ceqsalgALT  3226  alxfr  4869  copsex2t  4947  mosubopt  4962  ovmpt2df  6777  ov3  6782  tz7.48-1  7523  ac6c4  9288  fsum2dlem  14482  fprod2dlem  14691  gsum2d2lem  18353  padct  29471  exlimim  33160  exellim  33163  wl-lem-moexsb  33321  exlimddvf  33897  stoweidlem27  40007  fourierdlem31  40118  intsaluni  40310  isomenndlem  40507