Theorem rexlimd2 2622

Description: Version of rexlimd 2621 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 2578	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd2.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		r19.23t 2614	. . 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 147	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1484   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2490  df-rex 2491

This theorem is referenced by:  sbcrext  3078