Theorem rexxpf 4786

Description: Version of rexxp 4783 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 2732	. 2 |- ( E. x e. ( A X. B ) ph <-> E. v e. ( A X. B ) [ v / x ] ph )
2		cbvrexsv 2732	. . . 4 |- ( E. z e. B [ w / y ] ps <-> E. u e. B [ u / z ] [ w / y ] ps )
3	2	rexbii 2494	. . 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 1538	. . . 4 |- F/ w E. z e. B ps
5		nfcv 2329	. . . . 5 |- F/_ y B
6		nfs1v 1949	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfrexxy 2526	. . . 4 |- F/ y E. z e. B [ w / y ] ps
8		sbequ12 1781	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	rexbidv 2488	. . . 4 |- ( y = w -> ( E. z e. B ps <-> E. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvrex 2712	. . 3 |- ( E. y e. A E. z e. B ps <-> E. w e. A E. z e. B [ w / y ] ps )
11		vex 2752	. . . . . 6 |- w e. _V
12		vex 2752	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4255	. . . . 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 1956	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1956	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1599	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1956	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1949	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1599	. . . . . . 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 2798	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2752	. . . . . . . . . 10 |- y e. _V
26		vex 2752	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4249	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1781	. . . . . . . . . 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 1604	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1604	. . . . 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 4783	. . 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 1363   F/wnf 1470   E.wex 1502   [wsb 1772   E.wrex 2466   <.cop 3607   X. cxp 4636

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-iun 3900  df-opab 4077  df-xp 4644  df-rel 4645

This theorem is referenced by:  iunxpf  4787