Theorem ralxpf 4735

Description: Version of ralxp 4732 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
ralxpf.1	|- F/ y ph
ralxpf.2	|- F/ z ph
ralxpf.3	|- F/ x ps
ralxpf.4	|- ( x = <. y , z >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralxpf	|- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Distinct variable groups:   x, y, A   x, z, B, y

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   A( z)

Proof of Theorem ralxpf

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2694	. 2 |- ( A. x e. ( A X. B ) ph <-> A. v e. ( A X. B ) [ v / x ] ph )
2		cbvralsv 2694	. . . 4 |- ( A. z e. B [ w / y ] ps <-> A. u e. B [ u / z ] [ w / y ] ps )
3	2	ralbii 2463	. . 3 |- ( A. w e. A A. z e. B [ w / y ] ps <-> A. w e. A A. u e. B [ u / z ] [ w / y ] ps )
4		nfv 1508	. . . 4 |- F/ w A. z e. B ps
5		nfcv 2299	. . . . 5 |- F/_ y B
6		nfs1v 1919	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfralxy 2495	. . . 4 |- F/ y A. z e. B [ w / y ] ps
8		sbequ12 1751	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	ralbidv 2457	. . . 4 |- ( y = w -> ( A. z e. B ps <-> A. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvral 2676	. . 3 |- ( A. y e. A A. z e. B ps <-> A. w e. A A. z e. B [ w / y ] ps )
11		vex 2715	. . . . . 6 |- w e. _V
12		vex 2715	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4206	. . . . 5 |- ( v = <. w , u >. <-> E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) )
14		ralxpf.1	. . . . . . . 8 |- F/ y ph
15	14	nfsb 1926	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1926	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1569	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1926	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1919	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1569	. . . . . . 7 |- F/ z ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
22		ralxpf.3	. . . . . . . . 9 |- F/ x ps
23		ralxpf.4	. . . . . . . . 9 |- ( x = <. y , z >. -> ( ph <-> ps ) )
24	22, 23	sbhypf 2761	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2715	. . . . . . . . . 10 |- y e. _V
26		vex 2715	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4200	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1751	. . . . . . . . . 10 |- ( z = u -> ( [ w / y ] ps <-> [ u / z ] [ w / y ] ps ) )
29	8, 28	sylan9bb 458	. . . . . . . . 9 |- ( ( y = w /\ z = u ) -> ( ps <-> [ u / z ] [ w / y ] ps ) )
30	27, 29	sylbi 120	. . . . . . . 8 |- ( <. y , z >. = <. w , u >. -> ( ps <-> [ u / z ] [ w / y ] ps ) )
31	24, 30	sylan9bb 458	. . . . . . 7 |- ( ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
32	21, 31	exlimi 1574	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1574	. . . . 5 |- ( E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
34	13, 33	sylbi 120	. . . 4 |- ( v = <. w , u >. -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
35	34	ralxp 4732	. . 3 |- ( A. v e. ( A X. B ) [ v / x ] ph <-> A. w e. A A. u e. B [ u / z ] [ w / y ] ps )
36	3, 10, 35	3bitr4ri 212	. 2 |- ( A. v e. ( A X. B ) [ v / x ] ph <-> A. y e. A A. z e. B ps )
37	1, 36	bitri 183	1 |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   F/wnf 1440   E.wex 1472   [wsb 1742   A.wral 2435   <.cop 3564   X. cxp 4587

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-iun 3853  df-opab 4029  df-xp 4595  df-rel 4596

This theorem is referenced by: (None)