Theorem exlimih 1586

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)

Hypotheses

Ref	Expression
exlimih.1	⊢ (𝜓 → ∀𝑥𝜓)
exlimih.2	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
exlimih	⊢ (∃𝑥𝜑 → 𝜓)

Proof of Theorem exlimih

Step	Hyp	Ref	Expression
1		exlimih.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1491	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
3		exlimih.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	mpgbi 1445	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exlimi  1587  exlimiv  1591  19.43  1621  hbex  1629  ax6blem  1643  19.41h  1678  ax9o  1691  equid  1694  equsex  1721  cbvexh  1748  equs5a  1787  sb5rf  1845  equvin  1856  euan  2075  moexexdc  2103