Theorem rspcegf 42939

Description: A version of rspcev 3576 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
rspcegf.1	⊢ Ⅎ𝑥𝜓
rspcegf.2	⊢ Ⅎ𝑥𝐴
rspcegf.3	⊢ Ⅎ𝑥𝐵
rspcegf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspcegf	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Proof of Theorem rspcegf

Step	Hyp	Ref	Expression
1		rspcegf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
2		rspcegf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfel 2919	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
4		rspcegf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
6		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7		rspcegf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
9	1, 5, 8	spcegf 3546	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
10	9	anabsi5 667	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11		df-rex 3072	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
12	10, 11	sylibr 233	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1541  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2885  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-rex 3072  df-v 3445

This theorem is referenced by:  rspcef  42992  stoweidlem46  43975