Theorem exlimd2 1486

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1487 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)

Hypotheses

Ref	Expression
exlimd2.1	⊢ (𝜑 → ∀𝑥𝜑)
exlimd2.2	⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
exlimd2.3	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
exlimd2	⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Proof of Theorem exlimd2

Step	Hyp	Ref	Expression
1		exlimd2.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		exlimd2.2	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
3	1, 2	alrimih 1358	. 2 ⊢ (𝜑 → ∀𝑥(𝜒 → ∀𝑥𝜒))
4		exlimd2.3	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
5	1, 4	alrimih 1358	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
6		19.23ht 1386	. . 3 ⊢ (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓 → 𝜒) ↔ (∃𝑥𝜓 → 𝜒)))
7	6	biimpd 132	. 2 ⊢ (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → 𝜒)))
8	3, 5, 7	sylc 56	1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equsexd  1617  cbvexdh  1801