Theorem rexlimd2 2545

Description: Version of rexlimd 2544 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
rexlimd2.1	⊢ Ⅎ𝑥𝜑
rexlimd2.2	⊢ (𝜑 → Ⅎ𝑥𝜒)
rexlimd2.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
rexlimd2	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Proof of Theorem rexlimd2

Step	Hyp	Ref	Expression
1		rexlimd2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rexlimd2.3	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
3	1, 2	ralrimi 2501	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd2.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		r19.23t 2537	. . 3 ⊢ (Ⅎ𝑥𝜒 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)))
7	3, 6	mpbid 146	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1436   ∈ wcel 1480  ∀wral 2414  ∃wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420

This theorem is referenced by:  sbcrext  2981