Theorem exlimih 1573

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 1475	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
3		exlimih.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	mpgbi 1429	1 ⊢ (∃𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1330  ∃wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-gen 1426  ax-ie2 1471

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exlimi  1574  exlimiv  1578  19.43  1608  hbex  1616  ax6blem  1629  19.41h  1664  ax9o  1677  equid  1678  equsex  1707  cbvexh  1729  equs5a  1767  sb5rf  1825  equvin  1836  euan  2056  moexexdc  2084