Theorem exlimih 1641

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 1546	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
3		exlimih.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	mpgbi 1500	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1395  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1497  ax-ie2 1542

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exlimi  1642  exlimiv  1646  19.43  1676  hbex  1684  ax6blem  1698  19.41h  1733  ax9o  1746  equid  1749  equsex  1776  cbvexh  1803  equs5a  1842  sb5rf  1900  equvin  1911  euan  2136  moexexdc  2164