Theorem rspc2 2841

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

Hypotheses

Ref	Expression
rspc2.1	|- F/ x ch
rspc2.2	|- F/ y ps
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 2308	. . . 4 |- F/_ x D
2		rspc2.1	. . . 4 |- F/ x ch
3	1, 2	nfralxy 2504	. . 3 |- F/ x A. y e. D ch
4		rspc2.3	. . . 4 |- ( x = A -> ( ph <-> ch ) )
5	4	ralbidv 2466	. . 3 |- ( x = A -> ( A. y e. D ph <-> A. y e. D ch ) )
6	3, 5	rspc 2824	. 2 |- ( A e. C -> ( A. x e. C A. y e. D ph -> A. y e. D ch ) )
7		rspc2.2	. . 3 |- F/ y ps
8		rspc2.4	. . 3 |- ( y = B -> ( ch <-> ps ) )
9	7, 8	rspc 2824	. 2 |- ( B e. D -> ( A. y e. D ch -> ps ) )
10	6, 9	sylan9 407	1 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   F/wnf 1448   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728

This theorem is referenced by:  rspc2v  2843  disjiun  3977