Theorem exlimi 1485

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	nfri 1412	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		exlimi.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	exlimih 1484	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1349  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-gen 1338  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  19.36i  1562  euexex  1985  ceqsex  2592  sbhypf  2603  vtoclgf  2612  vtoclef  2626  copsexg  3981  copsex2g  3983  ralxpf  4482  rexxpf  4483  dmcoss  4601  fv3  5197  tz6.12c  5203  0neqopab  5550  bj-exlimmpi  9910