Theorem rspc2 2961

Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)

Hypotheses

Ref	Expression
rspc2.1	|- F/xch
rspc2.2	|- F/yps
rspc2.3	|- (x = A -> (ph <-> ch))
rspc2.4	|- (y = B -> (ch <-> ps))

Assertion

Ref	Expression
rspc2	|- ((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

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

Proof of Theorem rspc2

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 |- F/_xD
2		rspc2.1	. . . 4 |- F/xch
3	1, 2	nfral 2668	. . 3 |- F/xA.y e. D ch
4		rspc2.3	. . . 4 |- (x = A -> (ph <-> ch))
5	4	ralbidv 2635	. . 3 |- (x = A -> (A.y e. D ph <-> A.y e. D ch))
6	3, 5	rspc 2950	. 2 |- (A e. C -> (A.x e. C A.y e. D ph -> A.y e. D ch))
7		rspc2.2	. . 3 |- F/yps
8		rspc2.4	. . 3 |- (y = B -> (ch <-> ps))
9	7, 8	rspc 2950	. 2 |- (B e. D -> (A.y e. D ch -> ps))
10	6, 9	sylan9 638	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   F/wnf 1544   = 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:  rspc2v  2962