Theorem rspcegf 45713

Description: A version of rspcev 3580 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 2937	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
4		rspcegf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfan 1927	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
6		eleq1 2849	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7		rspcegf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
9	1, 5, 8	spcegf 3550	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
10	9	anabsi5 681	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11		df-rex 3088	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
12	10, 11	sylibr 237	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568  ∃wex 1807  Ⅎwnf 1811   ∈ wcel 2141  Ⅎwnfc 2908  ∃wrex 3087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rex 3088

This theorem is referenced by:  rspcef  45762  stoweidlem46  46730