Theorem csbopeq1a 6192

Description: Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3068). (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 2742	. . . . 5 |- x e. _V
2		vex 2742	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6153	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2183	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		csbeq1a 3068	. . 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 6152	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2183	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		csbeq1a 3068	. . 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 2211	1 |- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )

Syntax hints:   -> wi 4   = wceq 1353   [_csb 3059   <.cop 3597   ` cfv 5218   1stc1st 6142   2ndc2nd 6143

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-1st 6144  df-2nd 6145

This theorem is referenced by:  dfmpo  6227  f1od2  6239