Theorem csbopeq1a 6243

Description: Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3090). (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 2763	. . . . 5 |- x e. _V
2		vex 2763	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6204	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2199	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		csbeq1a 3090	. . 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 6203	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2199	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		csbeq1a 3090	. . 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 2227	1 |- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )

Syntax hints:   -> wi 4   = wceq 1364   [_csb 3081   <.cop 3622   ` cfv 5255   1stc1st 6193   2ndc2nd 6194

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by:  dfmpo  6278  f1od2  6290