Theorem exlimi 1803

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

Ref	Expression
exlimi.1	⊢ Ⅎxψ
exlimi.2	⊢ (φ → ψ)

Assertion

Ref	Expression
exlimi	⊢ (∃xφ → ψ)

Proof of Theorem exlimi

Step	Hyp	Ref	Expression
1		exlimi.1	. . 3 ⊢ Ⅎxψ
2	1	19.23 1801	. 2 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))
3		exlimi.2	. 2 ⊢ (φ → ψ)
4	2, 3	mpgbi 1549	1 ⊢ (∃xφ → ψ)

Syntax hints:   → wi 4  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  exlimih  1804  19.41  1879  equs5a  1887  sb5rf  2090  euan  2261  moexex  2273  ceqsex  2894  sbhypf  2905  vtoclgf  2914  vtoclef  2928  copsexg  4608  copsex2g  4610  ralxpf  4828  fv3  5342  tz6.12c  5348