Theorem csbopeq1a 6332

Description: Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3133). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
csbopeq1a	|- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )

Proof of Theorem csbopeq1a

Step	Hyp	Ref	Expression
1		vex 2802	. . . . 5 |- x e. _V
2		vex 2802	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6293	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2235	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		csbeq1a 3133	. . 3 |- ( y = ( 2nd ` A ) -> B = [_ ( 2nd ` A ) / y ]_ B )
6	4, 5	syl 14	. 2 |- ( A = <. x , y >. -> B = [_ ( 2nd ` A ) / y ]_ B )
7	1, 2	op1std 6292	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2235	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		csbeq1a 3133	. . 3 |- ( x = ( 1st ` A ) -> [_ ( 2nd ` A ) / y ]_ B = [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B )
10	8, 9	syl 14	. 2 |- ( A = <. x , y >. -> [_ ( 2nd ` A ) / y ]_ B = [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B )
11	6, 10	eqtr2d 2263	1 |- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )

Syntax hints:   -> wi 4   = wceq 1395   [_csb 3124   <.cop 3669   ` cfv 5317   1stc1st 6282   2ndc2nd 6283

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-1st 6284  df-2nd 6285

This theorem is referenced by:  dfmpo  6367  f1od2  6379