Theorem rspc2 2895

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 2350	. . . 4 |- F/_ x D
2		rspc2.1	. . . 4 |- F/ x ch
3	1, 2	nfralxy 2546	. . 3 |- F/ x A. y e. D ch
4		rspc2.3	. . . 4 |- ( x = A -> ( ph <-> ch ) )
5	4	ralbidv 2508	. . 3 |- ( x = A -> ( A. y e. D ph <-> A. y e. D ch ) )
6	3, 5	rspc 2878	. 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 2878	. 2 |- ( B e. D -> ( A. y e. D ch -> ps ) )
10	6, 9	sylan9 409	1 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1484   e. wcel 2178   A.wral 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778

This theorem is referenced by:  rspc2v  2897  disjiun  4054  dvmptfsum  15312