Theorem ralxpf 4981

Description: Version of ralxp 4976 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 3030	. 2 |- ( A. x e. ( A X. B ) ph <-> A. w e. ( A X. B ) [ w / x ] ph )
2		cbvralsv 3030	. . . 4 |- ( A. z e. B [ u / y ] ps <-> A. v e. B [ v / z ] [ u / y ] ps )
3	2	ralbii 2819	. . 3 |- ( A. u e. A A. z e. B [ u / y ] ps <-> A. u e. A A. v e. B [ v / z ] [ u / y ] ps )
4		nfv 1761	. . . 4 |- F/ u A. z e. B ps
5		nfcv 2592	. . . . 5 |- F/_ y B
6		nfs1v 2266	. . . . 5 |- F/ y [ u / y ] ps
7	5, 6	nfral 2774	. . . 4 |- F/ y A. z e. B [ u / y ] ps
8		sbequ12 2083	. . . . 5 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
9	8	ralbidv 2827	. . . 4 |- ( y = u -> ( A. z e. B ps <-> A. z e. B [ u / y ] ps ) )
10	4, 7, 9	cbvral 3015	. . 3 |- ( A. y e. A A. z e. B ps <-> A. u e. A A. z e. B [ u / y ] ps )
11		vex 3048	. . . . . 6 |- u e. _V
12		vex 3048	. . . . . 6 |- v e. _V
13	11, 12	eqvinop 4686	. . . . 5 |- ( w = <. u , v >. <-> E. y E. z ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) )
14		ralxpf.1	. . . . . . . 8 |- F/ y ph
15	14	nfsb 2269	. . . . . . 7 |- F/ y [ w / x ] ph
16	6	nfsb 2269	. . . . . . 7 |- F/ y [ v / z ] [ u / y ] ps
17	15, 16	nfbi 2017	. . . . . 6 |- F/ y ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 2269	. . . . . . . 8 |- F/ z [ w / x ] ph
20		nfs1v 2266	. . . . . . . 8 |- F/ z [ v / z ] [ u / y ] ps
21	19, 20	nfbi 2017	. . . . . . 7 |- F/ z ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps )
22		ralxpf.3	. . . . . . . . 9 |- F/ x ps
23		ralxpf.4	. . . . . . . . 9 |- ( x = <. y , z >. -> ( ph <-> ps ) )
24	22, 23	sbhypf 3095	. . . . . . . 8 |- ( w = <. y , z >. -> ( [ w / x ] ph <-> ps ) )
25		vex 3048	. . . . . . . . . 10 |- y e. _V
26		vex 3048	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4676	. . . . . . . . 9 |- ( <. y , z >. = <. u , v >. <-> ( y = u /\ z = v ) )
28		sbequ12 2083	. . . . . . . . . 10 |- ( z = v -> ( [ u / y ] ps <-> [ v / z ] [ u / y ] ps ) )
29	8, 28	sylan9bb 706	. . . . . . . . 9 |- ( ( y = u /\ z = v ) -> ( ps <-> [ v / z ] [ u / y ] ps ) )
30	27, 29	sylbi 199	. . . . . . . 8 |- ( <. y , z >. = <. u , v >. -> ( ps <-> [ v / z ] [ u / y ] ps ) )
31	24, 30	sylan9bb 706	. . . . . . 7 |- ( ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
32	21, 31	exlimi 1995	. . . . . 6 |- ( E. z ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
33	17, 32	exlimi 1995	. . . . 5 |- ( E. y E. z ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
34	13, 33	sylbi 199	. . . 4 |- ( w = <. u , v >. -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
35	34	ralxp 4976	. . 3 |- ( A. w e. ( A X. B ) [ w / x ] ph <-> A. u e. A A. v e. B [ v / z ] [ u / y ] ps )
36	3, 10, 35	3bitr4ri 282	. 2 |- ( A. w e. ( A X. B ) [ w / x ] ph <-> A. y e. A A. z e. B ps )
37	1, 36	bitri 253	1 |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   E.wex 1663   F/wnf 1667   [wsb 1797   A.wral 2737   <.cop 3974   X. cxp 4832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-iun 4280  df-opab 4462  df-xp 4840  df-rel 4841

This theorem is referenced by:  rexxpf  4982