Theorem opelopabsb 4142

Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
opelopabsb	|- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )

Distinct variable groups:   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x, y)   B( y)

Proof of Theorem opelopabsb

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

Step	Hyp	Ref	Expression
1		elopab 4140	. . . 4 |- ( <. A , B >. e. { <. u , v >. | [ u / x ] [ v / y ] ph } <-> E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) )
2		simpl 108	. . . . . . . 8 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> <. A , B >. = <. u , v >. )
3	2	eqcomd 2120	. . . . . . 7 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> <. u , v >. = <. A , B >. )
4		vex 2660	. . . . . . . 8 |- u e. _V
5		vex 2660	. . . . . . . 8 |- v e. _V
6	4, 5	opth 4119	. . . . . . 7 |- ( <. u , v >. = <. A , B >. <-> ( u = A /\ v = B ) )
7	3, 6	sylib 121	. . . . . 6 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> ( u = A /\ v = B ) )
8	7	2eximi 1563	. . . . 5 |- ( E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> E. u E. v ( u = A /\ v = B ) )
9		eeanv 1882	. . . . . 6 |- ( E. u E. v ( u = A /\ v = B ) <-> ( E. u u = A /\ E. v v = B ) )
10		isset 2663	. . . . . . 7 |- ( A e. _V <-> E. u u = A )
11		isset 2663	. . . . . . 7 |- ( B e. _V <-> E. v v = B )
12	10, 11	anbi12i 453	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) <-> ( E. u u = A /\ E. v v = B ) )
13	9, 12	bitr4i 186	. . . . 5 |- ( E. u E. v ( u = A /\ v = B ) <-> ( A e. _V /\ B e. _V ) )
14	8, 13	sylib 121	. . . 4 |- ( E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> ( A e. _V /\ B e. _V ) )
15	1, 14	sylbi 120	. . 3 |- ( <. A , B >. e. { <. u , v >. | [ u / x ] [ v / y ] ph } -> ( A e. _V /\ B e. _V ) )
16		nfv 1491	. . . 4 |- F/ u ph
17		nfv 1491	. . . 4 |- F/ v ph
18		nfs1v 1890	. . . 4 |- F/ x [ u / x ] [ v / y ] ph
19		nfs1v 1890	. . . . 5 |- F/ y [ v / y ] ph
20	19	nfsbxy 1893	. . . 4 |- F/ y [ u / x ] [ v / y ] ph
21		sbequ12 1727	. . . . 5 |- ( y = v -> ( ph <-> [ v / y ] ph ) )
22		sbequ12 1727	. . . . 5 |- ( x = u -> ( [ v / y ] ph <-> [ u / x ] [ v / y ] ph ) )
23	21, 22	sylan9bbr 456	. . . 4 |- ( ( x = u /\ y = v ) -> ( ph <-> [ u / x ] [ v / y ] ph ) )
24	16, 17, 18, 20, 23	cbvopab 3959	. . 3 |- { <. x , y >. | ph } = { <. u , v >. | [ u / x ] [ v / y ] ph }
25	15, 24	eleq2s 2209	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } -> ( A e. _V /\ B e. _V ) )
26		sbcex 2886	. . 3 |- ( [. A / x ]. [. B / y ]. ph -> A e. _V )
27		spesbc 2962	. . . 4 |- ( [. A / x ]. [. B / y ]. ph -> E. x [. B / y ]. ph )
28		sbcex 2886	. . . . 5 |- ( [. B / y ]. ph -> B e. _V )
29	28	exlimiv 1560	. . . 4 |- ( E. x [. B / y ]. ph -> B e. _V )
30	27, 29	syl 14	. . 3 |- ( [. A / x ]. [. B / y ]. ph -> B e. _V )
31	26, 30	jca 302	. 2 |- ( [. A / x ]. [. B / y ]. ph -> ( A e. _V /\ B e. _V ) )
32		opeq1 3671	. . . . 5 |- ( z = A -> <. z , w >. = <. A , w >. )
33	32	eleq1d 2183	. . . 4 |- ( z = A -> ( <. z , w >. e. { <. x , y >. | ph } <-> <. A , w >. e. { <. x , y >. | ph } ) )
34		dfsbcq2 2881	. . . 4 |- ( z = A -> ( [ z / x ] [ w / y ] ph <-> [. A / x ]. [ w / y ] ph ) )
35	33, 34	bibi12d 234	. . 3 |- ( z = A -> ( ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) <-> ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) ) )
36		opeq2 3672	. . . . 5 |- ( w = B -> <. A , w >. = <. A , B >. )
37	36	eleq1d 2183	. . . 4 |- ( w = B -> ( <. A , w >. e. { <. x , y >. | ph } <-> <. A , B >. e. { <. x , y >. | ph } ) )
38		dfsbcq2 2881	. . . . 5 |- ( w = B -> ( [ w / y ] ph <-> [. B / y ]. ph ) )
39	38	sbcbidv 2935	. . . 4 |- ( w = B -> ( [. A / x ]. [ w / y ] ph <-> [. A / x ]. [. B / y ]. ph ) )
40	37, 39	bibi12d 234	. . 3 |- ( w = B -> ( ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) <-> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) ) )
41		nfopab1 3957	. . . . . 6 |- F/_ x { <. x , y >. | ph }
42	41	nfel2 2268	. . . . 5 |- F/ x <. z , w >. e. { <. x , y >. | ph }
43		nfs1v 1890	. . . . 5 |- F/ x [ z / x ] [ w / y ] ph
44	42, 43	nfbi 1551	. . . 4 |- F/ x ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph )
45		opeq1 3671	. . . . . 6 |- ( x = z -> <. x , w >. = <. z , w >. )
46	45	eleq1d 2183	. . . . 5 |- ( x = z -> ( <. x , w >. e. { <. x , y >. | ph } <-> <. z , w >. e. { <. x , y >. | ph } ) )
47		sbequ12 1727	. . . . 5 |- ( x = z -> ( [ w / y ] ph <-> [ z / x ] [ w / y ] ph ) )
48	46, 47	bibi12d 234	. . . 4 |- ( x = z -> ( ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph ) <-> ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) ) )
49		nfopab2 3958	. . . . . . 7 |- F/_ y { <. x , y >. | ph }
50	49	nfel2 2268	. . . . . 6 |- F/ y <. x , w >. e. { <. x , y >. | ph }
51		nfs1v 1890	. . . . . 6 |- F/ y [ w / y ] ph
52	50, 51	nfbi 1551	. . . . 5 |- F/ y ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph )
53		opeq2 3672	. . . . . . 7 |- ( y = w -> <. x , y >. = <. x , w >. )
54	53	eleq1d 2183	. . . . . 6 |- ( y = w -> ( <. x , y >. e. { <. x , y >. | ph } <-> <. x , w >. e. { <. x , y >. | ph } ) )
55		sbequ12 1727	. . . . . 6 |- ( y = w -> ( ph <-> [ w / y ] ph ) )
56	54, 55	bibi12d 234	. . . . 5 |- ( y = w -> ( ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) <-> ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph ) ) )
57		opabid 4139	. . . . 5 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
58	52, 56, 57	chvar 1713	. . . 4 |- ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph )
59	44, 48, 58	chvar 1713	. . 3 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph )
60	35, 40, 59	vtocl2g 2721	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) )
61	25, 31, 60	pm5.21nii 676	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1314   E.wex 1451   e. wcel 1463   [wsb 1718   _Vcvv 2657   [.wsbc 2878   <.cop 3496   {copab 3948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-opab 3950

This theorem is referenced by:  brabsb  4143  opelopabaf  4155  opelopabf  4156  difopab  4632  isarep1  5167