Theorem rspc2 3568

Description: Restricted specialization with two quantifiers, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

Hypotheses

Ref	Expression
rspc2.1	⊢ Ⅎ𝑥𝜒
rspc2.2	⊢ Ⅎ𝑦𝜓
rspc2.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
rspc2.4	⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
rspc2	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem rspc2

Step	Hyp	Ref	Expression
1		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥𝐷
2		rspc2.1	. . . 4 ⊢ Ⅎ𝑥𝜒
3	1, 2	nfralw 3151	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒
4		rspc2.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
5	4	ralbidv 3122	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒))
6	3, 5	rspc 3547	. 2 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 𝜒))
7		rspc2.2	. . 3 ⊢ Ⅎ𝑦𝜓
8		rspc2.4	. . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓))
9	7, 8	rspc 3547	. 2 ⊢ (𝐵 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 𝜒 → 𝜓))
10	6, 9	sylan9 507	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1541  Ⅎwnf 1789   ∈ wcel 2109  ∀wral 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-v 3432

This theorem is referenced by:  reu2eqd  3674  reuop  6193  fvmpocurryd  8071  dvmptfsum  25120  poimirlem26  35782  fphpd  40618  reupr  44926