Theorem ralxpf 5138

Description: Version of ralxp 5133 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 3092	. 2 |- ( A. x e. ( A X. B ) ph <-> A. w e. ( A X. B ) [ w / x ] ph )
2		cbvralsv 3092	. . . 4 |- ( A. z e. B [ u / y ] ps <-> A. v e. B [ v / z ] [ u / y ] ps )
3	2	ralbii 2885	. . 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 1712	. . . 4 |- F/ u A. z e. B ps
5		nfcv 2616	. . . . 5 |- F/_ y B
6		nfs1v 2183	. . . . 5 |- F/ y [ u / y ] ps
7	5, 6	nfral 2840	. . . 4 |- F/ y A. z e. B [ u / y ] ps
8		sbequ12 1997	. . . . 5 |- ( y = u -> ( ps <-> [ u / y ] ps ) )
9	8	ralbidv 2893	. . . 4 |- ( y = u -> ( A. z e. B ps <-> A. z e. B [ u / y ] ps ) )
10	4, 7, 9	cbvral 3077	. . 3 |- ( A. y e. A A. z e. B ps <-> A. u e. A A. z e. B [ u / y ] ps )
11		vex 3109	. . . . . 6 |- u e. _V
12		vex 3109	. . . . . 6 |- v e. _V
13	11, 12	eqvinop 4721	. . . . 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 2186	. . . . . . 7 |- F/ y [ w / x ] ph
16	6	nfsb 2186	. . . . . . 7 |- F/ y [ v / z ] [ u / y ] ps
17	15, 16	nfbi 1939	. . . . . 6 |- F/ y ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 2186	. . . . . . . 8 |- F/ z [ w / x ] ph
20		nfs1v 2183	. . . . . . . 8 |- F/ z [ v / z ] [ u / y ] ps
21	19, 20	nfbi 1939	. . . . . . 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 3153	. . . . . . . 8 |- ( w = <. y , z >. -> ( [ w / x ] ph <-> ps ) )
25		vex 3109	. . . . . . . . . 10 |- y e. _V
26		vex 3109	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4711	. . . . . . . . 9 |- ( <. y , z >. = <. u , v >. <-> ( y = u /\ z = v ) )
28		sbequ12 1997	. . . . . . . . . 10 |- ( z = v -> ( [ u / y ] ps <-> [ v / z ] [ u / y ] ps ) )
29	8, 28	sylan9bb 697	. . . . . . . . 9 |- ( ( y = u /\ z = v ) -> ( ps <-> [ v / z ] [ u / y ] ps ) )
30	27, 29	sylbi 195	. . . . . . . 8 |- ( <. y , z >. = <. u , v >. -> ( ps <-> [ v / z ] [ u / y ] ps ) )
31	24, 30	sylan9bb 697	. . . . . . 7 |- ( ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
32	21, 31	exlimi 1917	. . . . . 6 |- ( E. z ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
33	17, 32	exlimi 1917	. . . . 5 |- ( E. y E. z ( w = <. y , z >. /\ <. y , z >. = <. u , v >. ) -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
34	13, 33	sylbi 195	. . . 4 |- ( w = <. u , v >. -> ( [ w / x ] ph <-> [ v / z ] [ u / y ] ps ) )
35	34	ralxp 5133	. . 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 278	. 2 |- ( A. w e. ( A X. B ) [ w / x ] ph <-> A. y e. A A. z e. B ps )
37	1, 36	bitri 249	1 |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   E.wex 1617   F/wnf 1621   [wsb 1744   A.wral 2804   <.cop 4022   X. cxp 4986

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-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  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-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-iun 4317  df-opab 4498  df-xp 4994  df-rel 4995

This theorem is referenced by:  rexxpf  5139