Theorem rspc2 3215

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 2616	. . . 4 |- F/_ x D
2		rspc2.1	. . . 4 |- F/ x ch
3	1, 2	nfral 2840	. . 3 |- F/ x A. y e. D ch
4		rspc2.3	. . . 4 |- ( x = A -> ( ph <-> ch ) )
5	4	ralbidv 2893	. . 3 |- ( x = A -> ( A. y e. D ph <-> A. y e. D ch ) )
6	3, 5	rspc 3201	. 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 3201	. 2 |- ( B e. D -> ( A. y e. D ch -> ps ) )
10	6, 9	sylan9 655	1 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   F/wnf 1621   e. wcel 1823   A.wral 2804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108

This theorem is referenced by:  rspc2v  3216  reu2eqd  3293  fvmpt2curryd  6992  dvmptfsum  22542  fphpd  30989