Theorem ralxpf 4482

Description: Version of ralxp 4479 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 2544	. 2 |- ( A. x e. ( A X. B ) ph <-> A. v e. ( A X. B ) [ v / x ] ph )
2		cbvralsv 2544	. . . 4 |- ( A. z e. B [ w / y ] ps <-> A. u e. B [ u / z ] [ w / y ] ps )
3	2	ralbii 2330	. . 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 1421	. . . 4 |- F/ w A. z e. B ps
5		nfcv 2178	. . . . 5 |- F/_ y B
6		nfs1v 1815	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfralxy 2360	. . . 4 |- F/ y A. z e. B [ w / y ] ps
8		sbequ12 1654	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	ralbidv 2326	. . . 4 |- ( y = w -> ( A. z e. B ps <-> A. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvral 2529	. . 3 |- ( A. y e. A A. z e. B ps <-> A. w e. A A. z e. B [ w / y ] ps )
11		vex 2560	. . . . . 6 |- w e. _V
12		vex 2560	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 3980	. . . . 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 1822	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1822	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1481	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1822	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1815	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1481	. . . . . . 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 2603	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2560	. . . . . . . . . 10 |- y e. _V
26		vex 2560	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 3974	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1654	. . . . . . . . . 10 |- ( z = u -> ( [ w / y ] ps <-> [ u / z ] [ w / y ] ps ) )
29	8, 28	sylan9bb 435	. . . . . . . . 9 |- ( ( y = w /\ z = u ) -> ( ps <-> [ u / z ] [ w / y ] ps ) )
30	27, 29	sylbi 114	. . . . . . . 8 |- ( <. y , z >. = <. w , u >. -> ( ps <-> [ u / z ] [ w / y ] ps ) )
31	24, 30	sylan9bb 435	. . . . . . 7 |- ( ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
32	21, 31	exlimi 1485	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1485	. . . . 5 |- ( E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
34	13, 33	sylbi 114	. . . 4 |- ( v = <. w , u >. -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
35	34	ralxp 4479	. . 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 202	. 2 |- ( A. v e. ( A X. B ) [ v / x ] ph <-> A. y e. A A. z e. B ps )
37	1, 36	bitri 173	1 |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   E.wex 1381   [wsb 1645   A.wral 2306   <.cop 3378   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-xp 4351  df-rel 4352

This theorem is referenced by: (None)