Theorem rspc2v 2961

Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)

Hypotheses

Ref	Expression
rspc2v.1	⊢ (x = A → (φ ↔ χ))
rspc2v.2	⊢ (y = B → (χ ↔ ψ))

Assertion

Ref	Expression
rspc2v	⊢ ((A ∈ C ∧ B ∈ D) → (∀x ∈ C ∀y ∈ D φ → ψ))

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

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

Proof of Theorem rspc2v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxχ
2		nfv 1619	. 2 ⊢ Ⅎyψ
3		rspc2v.1	. 2 ⊢ (x = A → (φ ↔ χ))
4		rspc2v.2	. 2 ⊢ (y = B → (χ ↔ ψ))
5	1, 2, 3, 4	rspc2 2960	1 ⊢ ((A ∈ C ∧ B ∈ D) → (∀x ∈ C ∀y ∈ D φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614

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 2478  df-ral 2619  df-v 2861

This theorem is referenced by:  rspc2va  2962  rspc3v  2964  ncfinraise  4481  nnpweq  4523  isorel  5489  isotr  5495  fovcl  5588  caovcld  5622  caovcomg  5624  extd  5923  symd  5924  antid  5929  connexd  5931