Theorem exlimd 2252

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 2250	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2239	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 253	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∃wex 1798  Ⅎwnf 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-ex 1799  df-nf 1803

This theorem is referenced by:  exlimimdd  2253  exlimdh  2323  equs5  2490  moexexlem  2652  2eu6  2682  ceqsalgALT  3489  alxfr  5363  copsex2t  5460  mosubopt  5478  ov3  7555  tz7.48-1  8409  ac6c4  10435  fsum2dlem  15780  fprod2dlem  15993  gsum2d2lem  19996  exlimim  37800  exellim  37802  wl-lem-moexsb  38035  exlimddvf  38584  mnringmulrcld  44768  fourierdlem31  46676  or2expropbi  47592  ich2exprop  48041  ichreuopeq  48043  reuopreuprim  48096