Theorem rspc2 3125

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 2592	. . . 4 |- F/_ x D
2		rspc2.1	. . . 4 |- F/ x ch
3	1, 2	nfral 2769	. . 3 |- F/ x A. y e. D ch
4		rspc2.3	. . . 4 |- ( x = A -> ( ph <-> ch ) )
5	4	ralbidv 2809	. . 3 |- ( x = A -> ( A. y e. D ph <-> A. y e. D ch ) )
6	3, 5	rspc 3111	. 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 3111	. 2 |- ( B e. D -> ( A. y e. D ch -> ps ) )
10	6, 9	sylan9 667	1 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1447   F/wnf 1670   e. wcel 1890   A.wral 2736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741  df-v 3014

This theorem is referenced by:  rspc2v  3126  reu2eqd  3202  fvmpt2curryd  7004  dvmptfsum  22938  poimirlem26  31967  fphpd  35660