Theorem rspcegf 42455

Description: A version of rspcev 3552 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 2920	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
4		rspcegf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
6		eleq1 2826	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7		rspcegf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
9	1, 5, 8	spcegf 3521	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
10	9	anabsi5 665	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11		df-rex 3069	. 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 395   = wceq 1539  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rex 3069  df-v 3424

This theorem is referenced by:  rspcef  42509  stoweidlem46  43477