Theorem sbcopeq1a 6093

Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2922 that avoids the existential quantifiers of copsexg 4174). (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 2692	. . . . 5 |- x e. _V
2		vex 2692	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6055	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2146	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		sbceq1a 2922	. . 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 6054	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2146	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		sbceq1a 2922	. . 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 188	1 |- ( A = <. x , y >. -> ( [. ( 1st ` A ) / x ]. [. ( 2nd ` A ) / y ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   [.wsbc 2913   <.cop 3535   ` cfv 5131   1stc1st 6044   2ndc2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by:  dfopab2  6095  dfoprab3s  6096