Theorem exlimd 2242

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 2240	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2227	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	syl6ib 241	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1851  Ⅎwnf 1855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202

This theorem depends on definitions:  df-bi 197  df-ex 1852  df-nf 1857

This theorem is referenced by:  exlimdd  2243  exlimdh  2313  equs5  2496  moexex  2689  2eu6  2706  exists2  2710  ceqsalgALT  3380  alxfr  5006  copsex2t  5084  mosubopt  5103  ovmpt2df  6938  ov3  6943  tz7.48-1  7690  ac6c4  9504  fsum2dlem  14708  fprod2dlem  14916  gsum2d2lem  18578  padct  29831  exlimim  33519  exellim  33522  wl-lem-moexsb  33677  exlimddvf  34251  stoweidlem27  40755  fourierdlem31  40866  intsaluni  41058  isomenndlem  41258