Theorem rexxpf 4824

Description: Version of rexxp 4821 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 2754	. 2 |- ( E. x e. ( A X. B ) ph <-> E. v e. ( A X. B ) [ v / x ] ph )
2		cbvrexsv 2754	. . . 4 |- ( E. z e. B [ w / y ] ps <-> E. u e. B [ u / z ] [ w / y ] ps )
3	2	rexbii 2512	. . 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 1550	. . . 4 |- F/ w E. z e. B ps
5		nfcv 2347	. . . . 5 |- F/_ y B
6		nfs1v 1966	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfrexw 2544	. . . 4 |- F/ y E. z e. B [ w / y ] ps
8		sbequ12 1793	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	rexbidv 2506	. . . 4 |- ( y = w -> ( E. z e. B ps <-> E. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvrex 2734	. . 3 |- ( E. y e. A E. z e. B ps <-> E. w e. A E. z e. B [ w / y ] ps )
11		vex 2774	. . . . . 6 |- w e. _V
12		vex 2774	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4286	. . . . 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 1973	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1973	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1611	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1973	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1966	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1611	. . . . . . 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 2821	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2774	. . . . . . . . . 10 |- y e. _V
26		vex 2774	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4280	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1793	. . . . . . . . . 10 |- ( z = u -> ( [ w / y ] ps <-> [ u / z ] [ w / y ] ps ) )
29	8, 28	sylan9bb 462	. . . . . . . . 9 |- ( ( y = w /\ z = u ) -> ( ps <-> [ u / z ] [ w / y ] ps ) )
30	27, 29	sylbi 121	. . . . . . . 8 |- ( <. y , z >. = <. w , u >. -> ( ps <-> [ u / z ] [ w / y ] ps ) )
31	24, 30	sylan9bb 462	. . . . . . 7 |- ( ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
32	21, 31	exlimi 1616	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1616	. . . . 5 |- ( E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
34	13, 33	sylbi 121	. . . 4 |- ( v = <. w , u >. -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
35	34	rexxp 4821	. . 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 213	. 2 |- ( E. v e. ( A X. B ) [ v / x ] ph <-> E. y e. A E. z e. B ps )
37	1, 36	bitri 184	1 |- ( E. x e. ( A X. B ) ph <-> E. y e. A E. z e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   F/wnf 1482   E.wex 1514   [wsb 1784   E.wrex 2484   <.cop 3635   X. cxp 4672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-iun 3928  df-opab 4105  df-xp 4680  df-rel 4681

This theorem is referenced by:  iunxpf  4825