Theorem exlimd 2256

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 2254	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
5	4	19.9 2243	. 2 ⊢ (∃𝑥𝜒 ↔ 𝜒)
6	3, 5	imbitrdi 254	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  exlimimdd  2257  exlimdh  2325  equs5  2491  moexexlem  2653  2eu6  2683  ceqsalgALT  3489  alxfr  5376  copsex2t  5473  mosubopt  5491  ov3  7580  tz7.48-1  8436  ac6c4  10487  fsum2dlem  15860  fprod2dlem  16073  gsum2d2lem  20106  exlimim  38104  exellim  38106  wl-lem-moexsb  38339  exlimddvf  38877  mnringmulrcld  45074  fourierdlem31  46974  or2expropbi  47930  ich2exprop  48379  ichreuopeq  48381  reuopreuprim  48434