Theorem sbcopeq1a 5813

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2773 that avoids the existential quantifiers of copsexg 3981). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
sbcopeq1a	|- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Proof of Theorem sbcopeq1a

Step	Hyp	Ref	Expression
1		vex 2560	. . . . 5 |- x e. _V
2		vex 2560	. . . . 5 |- y e. _V
3	1, 2	op2ndd 5776	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2045	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		sbceq1a 2773	. . 3 |- ( y = ( 2nd ` A ) -> ( ph <-> [. ( 2nd ` A ) / y ]. ph ) )
6	4, 5	syl 14	. 2 |- ( A = <. x , y >. -> ( ph <-> [. ( 2nd ` A ) / y ]. ph ) )
7	1, 2	op1std 5775	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2045	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		sbceq1a 2773	. . 3 |- ( x = ( 1st ` A ) -> ( [. ( 2nd ` A ) / y ]. ph <-> [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph ) )
10	8, 9	syl 14	. 2 |- ( A = <. x , y >. -> ( [. ( 2nd ` A ) / y ]. ph <-> [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph ) )
11	6, 10	bitr2d 178	1 |- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   [.wsbc 2764   <.cop 3378   ` cfv 4902   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by:  dfopab2  5815  dfoprab3s  5816