Theorem rexlimd2 3260

Description: Version of rexlimd 3261 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 3252	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
4		rexlimd2.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		r19.23t 3250	. . 3 ⊢ (Ⅎ𝑥𝜒 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)))
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)))
7	3, 6	mpbid 231	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-nf 1778  df-ral 3059  df-rex 3068

This theorem is referenced by:  rexlimd  3261  sbcrext  3868