Theorem exlimih 1646

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 1551	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
3		exlimih.2	. 2 ⊢ (𝜑 → 𝜓)
4	2, 3	mpgbi 1505	1 ⊢ (∃𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-gen 1502  ax-ie2 1547

This proof depends on definitions:  df-bi 117

This theorem is used by:  exlimi  1647  exlimiv  1651  19.43  1681  hbex  1689  hbn  1703  19.41h  1737  ax9o  1750  equid  1753  equsex  1780  cbvexh  1808  equs5a  1847  sb5rf  1905  equvin  1916  euan  2143  moexexdc  2171