Theorem rspc2 3320

Description: Restricted specialization with two quantifiers, 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 2764	. . . 4 |- F/_ x D
2		rspc2.1	. . . 4 |- F/ x ch
3	1, 2	nfral 2945	. . 3 |- F/ x A. y e. D ch
4		rspc2.3	. . . 4 |- ( x = A -> ( ph <-> ch ) )
5	4	ralbidv 2986	. . 3 |- ( x = A -> ( A. y e. D ph <-> A. y e. D ch ) )
6	3, 5	rspc 3303	. 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 3303	. 2 |- ( B e. D -> ( A. y e. D ch -> ps ) )
10	6, 9	sylan9 689	1 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  rspc2v  3322  reu2eqd  3403  fvmpt2curryd  7397  dvmptfsum  23738  poimirlem26  33435  fphpd  37380