Theorem exlimd 1650

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 1572	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		exlimd.2	. . 3 ⊢ Ⅎ𝑥𝜒
4	3	nfri 1572	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
5		exlimd.3	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
6	2, 4, 5	exlimdh 1649	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1513  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  exlimdd  1925  ceqsalg  2850  copsex2t  4385  alxfr  4607  mosubopt  4840  ovmpodf  6220  ovi3  6226  fsum2dlemstep  12203  fprod2dlemstep  12391  lss1d  14722  bj-exlimmp  16809