Theorem exlimd 1585

Description: Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-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	1	nfri 1507	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
4	3	nfri 1507	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
5		exlimd.3	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 4, 5	exlimdh 1584	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  Ⅎwnf 1448  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  exlimdd  1860  ceqsalg  2754  copsex2t  4223  alxfr  4439  mosubopt  4669  ovmpodf  5973  ovi3  5978  fsum2dlemstep  11375  fprod2dlemstep  11563  bj-exlimmp  13650