Theorem rspc2va 2963

Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)

Hypotheses

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

Assertion

Ref	Expression
rspc2va	|- (((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 rspc2va

Step	Hyp	Ref	Expression
1		rspc2v.1	. . 3 |- (x = A -> (ph <-> ch))
2		rspc2v.2	. . 3 |- (y = B -> (ch <-> ps))
3	1, 2	rspc2v 2962	. 2 |- ((A e. C /\ B e. D) -> (A.x e. C A.y e. D ph -> ps))
4	3	imp 418	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 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:  isocnv  5492