Theorem rspc2 2961

Description: 2-variable restricted specialization, 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 2490	. . . 4 ⊢ ℲxD
2		rspc2.1	. . . 4 ⊢ Ⅎxχ
3	1, 2	nfral 2668	. . 3 ⊢ Ⅎx∀y ∈ D χ
4		rspc2.3	. . . 4 ⊢ (x = A → (φ ↔ χ))
5	4	ralbidv 2635	. . 3 ⊢ (x = A → (∀y ∈ D φ ↔ ∀y ∈ D χ))
6	3, 5	rspc 2950	. 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 2950	. 2 ⊢ (B ∈ D → (∀y ∈ D χ → ψ))
10	6, 9	sylan9 638	1 ⊢ ((A ∈ C ∧ B ∈ D) → (∀x ∈ C ∀y ∈ D φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862

This theorem is referenced by:  rspc2v  2962