Theorem rexxpf 4501

Description: Version of rexxp 4498 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (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
rexxpf	|- ( E. x e. ( A X. B ) ph <-> E. y e. A E. 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 rexxpf

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

Step	Hyp	Ref	Expression
1		cbvrexsv 2589	. 2 |- ( E. x e. ( A X. B ) ph <-> E. v e. ( A X. B ) [ v / x ] ph )
2		cbvrexsv 2589	. . . 4 |- ( E. z e. B [ w / y ] ps <-> E. u e. B [ u / z ] [ w / y ] ps )
3	2	rexbii 2373	. . 3 |- ( E. w e. A E. z e. B [ w / y ] ps <-> E. w e. A E. u e. B [ u / z ] [ w / y ] ps )
4		nfv 1461	. . . 4 |- F/ w E. z e. B ps
5		nfcv 2219	. . . . 5 |- F/_ y B
6		nfs1v 1856	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfrexxy 2403	. . . 4 |- F/ y E. z e. B [ w / y ] ps
8		sbequ12 1694	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	rexbidv 2369	. . . 4 |- ( y = w -> ( E. z e. B ps <-> E. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvrex 2574	. . 3 |- ( E. y e. A E. z e. B ps <-> E. w e. A E. z e. B [ w / y ] ps )
11		vex 2604	. . . . . 6 |- w e. _V
12		vex 2604	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 3998	. . . . 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 1863	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1863	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1521	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1863	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1856	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1521	. . . . . . 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 2648	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2604	. . . . . . . . . 10 |- y e. _V
26		vex 2604	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 3992	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1694	. . . . . . . . . 10 |- ( z = u -> ( [ w / y ] ps <-> [ u / z ] [ w / y ] ps ) )
29	8, 28	sylan9bb 449	. . . . . . . . 9 |- ( ( y = w /\ z = u ) -> ( ps <-> [ u / z ] [ w / y ] ps ) )
30	27, 29	sylbi 119	. . . . . . . 8 |- ( <. y , z >. = <. w , u >. -> ( ps <-> [ u / z ] [ w / y ] ps ) )
31	24, 30	sylan9bb 449	. . . . . . 7 |- ( ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
32	21, 31	exlimi 1525	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1525	. . . . 5 |- ( E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
34	13, 33	sylbi 119	. . . 4 |- ( v = <. w , u >. -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
35	34	rexxp 4498	. . 3 |- ( E. v e. ( A X. B ) [ v / x ] ph <-> E. w e. A E. u e. B [ u / z ] [ w / y ] ps )
36	3, 10, 35	3bitr4ri 211	. 2 |- ( E. v e. ( A X. B ) [ v / x ] ph <-> E. y e. A E. z e. B ps )
37	1, 36	bitri 182	1 |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   E.wex 1421   [wsb 1685   E.wrex 2349   <.cop 3401   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by:  iunxpf  4502