Theorem rexxpf 4883

Description: Version of rexxp 4880 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 2785	. 2 |- ( E. x e. ( A X. B ) ph <-> E. v e. ( A X. B ) [ v / x ] ph )
2		cbvrexsv 2785	. . . 4 |- ( E. z e. B [ w / y ] ps <-> E. u e. B [ u / z ] [ w / y ] ps )
3	2	rexbii 2540	. . 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 1577	. . . 4 |- F/ w E. z e. B ps
5		nfcv 2375	. . . . 5 |- F/_ y B
6		nfs1v 1992	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfrexw 2572	. . . 4 |- F/ y E. z e. B [ w / y ] ps
8		sbequ12 1819	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	rexbidv 2534	. . . 4 |- ( y = w -> ( E. z e. B ps <-> E. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvrex 2765	. . 3 |- ( E. y e. A E. z e. B ps <-> E. w e. A E. z e. B [ w / y ] ps )
11		vex 2806	. . . . . 6 |- w e. _V
12		vex 2806	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4341	. . . . 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 1999	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1999	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1638	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1999	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1992	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1638	. . . . . . 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 2854	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2806	. . . . . . . . . 10 |- y e. _V
26		vex 2806	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4335	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1819	. . . . . . . . . 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 1643	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1643	. . . . 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 4880	. . 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 1398   F/wnf 1509   E.wex 1541   [wsb 1810   E.wrex 2512   <.cop 3676   X. cxp 4729

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-iun 3977  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  iunxpf  4884