Theorem rspc2v 2961

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

Hypotheses

Ref	Expression
rspc2v.1	|- (x = A -> (ph <-> ch))
rspc2v.2	|- (y = B -> (ch <-> ps))

Assertion

Ref	Expression
rspc2v	|- ((A e. C /\ B e. D) -> (A.x e. C A.y e. D ph -> ps))

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

Allowed substitution hints:   ph(x,y)   ps(x)   ch(y)   B(x)   C(y)

Proof of Theorem rspc2v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xch
2		nfv 1619	. 2 |- F/yps
3		rspc2v.1	. 2 |- (x = A -> (ph <-> ch))
4		rspc2v.2	. 2 |- (y = B -> (ch <-> ps))
5	1, 2, 3, 4	rspc2 2960	1 |- ((A e. C /\ B e. D) -> (A.x e. C A.y e. D ph -> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.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