Theorem rexxpf 4868

Description: Version of rexxp 4865 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 2782	. 2 |- ( E. x e. ( A X. B ) ph <-> E. v e. ( A X. B ) [ v / x ] ph )
2		cbvrexsv 2782	. . . 4 |- ( E. z e. B [ w / y ] ps <-> E. u e. B [ u / z ] [ w / y ] ps )
3	2	rexbii 2537	. . 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 1574	. . . 4 |- F/ w E. z e. B ps
5		nfcv 2372	. . . . 5 |- F/_ y B
6		nfs1v 1990	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfrexw 2569	. . . 4 |- F/ y E. z e. B [ w / y ] ps
8		sbequ12 1817	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	rexbidv 2531	. . . 4 |- ( y = w -> ( E. z e. B ps <-> E. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvrex 2762	. . 3 |- ( E. y e. A E. z e. B ps <-> E. w e. A E. z e. B [ w / y ] ps )
11		vex 2802	. . . . . 6 |- w e. _V
12		vex 2802	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4328	. . . . 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 1997	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1997	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1635	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1997	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1990	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1635	. . . . . . 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 2850	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2802	. . . . . . . . . 10 |- y e. _V
26		vex 2802	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4322	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1817	. . . . . . . . . 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 1640	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1640	. . . . 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 4865	. . 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 1395   F/wnf 1506   E.wex 1538   [wsb 1808   E.wrex 2509   <.cop 3669   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iun 3966  df-opab 4145  df-xp 4724  df-rel 4725

This theorem is referenced by:  iunxpf  4869