Theorem exlimdh 1589

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)

Hypotheses

Ref	Expression
exlimdh.1	⊢ (𝜑 → ∀𝑥𝜑)
exlimdh.2	⊢ (𝜒 → ∀𝑥𝜒)
exlimdh.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimdh	⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Proof of Theorem exlimdh

Step	Hyp	Ref	Expression
1		exlimdh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		exlimdh.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	alrimih 1462	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
4		exlimdh.2	. . 3 ⊢ (𝜒 → ∀𝑥𝜒)
5	4	19.23h 1491	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) ↔ (∃𝑥𝜓 → 𝜒))
6	3, 5	sylib 121	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

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-5 1440  ax-gen 1442  ax-ie2 1487

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exlimd  1590  exim  1592  exlimdv  1812  equs5  1822  cbvexdh  1919  exists2  2116