Theorem rspc2 3631

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 2903	. . . 4 ⊢ Ⅎ𝑥𝐷
2		rspc2.1	. . . 4 ⊢ Ⅎ𝑥𝜒
3	1, 2	nfralw 3309	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒
4		rspc2.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))
5	4	ralbidv 3176	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒))
6	3, 5	rspc 3610	. 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 3610	. 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 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060

This theorem is referenced by:  reu2eqd  3745  reuop  6315  fvmpocurryd  8295  dvmptfsum  26028  poimirlem26  37633  fphpd  42804  reupr  47447