Theorem csbopeq1a 6167

Description: Equality theorem for substitution of a class A for an ordered pair <. x , y >. in B (analog of csbeq1a 3058). (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 2733	. . . . 5 |- x e. _V
2		vex 2733	. . . . 5 |- y e. _V
3	1, 2	op2ndd 6128	. . . 4 |- ( A = <. x , y >. -> ( 2nd ` A ) = y )
4	3	eqcomd 2176	. . 3 |- ( A = <. x , y >. -> y = ( 2nd ` A ) )
5		csbeq1a 3058	. . 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 6127	. . . 4 |- ( A = <. x , y >. -> ( 1st ` A ) = x )
8	7	eqcomd 2176	. . 3 |- ( A = <. x , y >. -> x = ( 1st ` A ) )
9		csbeq1a 3058	. . 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 2204	1 |- ( A = <. x , y >. -> [_ ( 1st ` A ) / x ]_ [_ ( 2nd ` A ) / y ]_ B = B )

Syntax hints:   -> wi 4   = wceq 1348   [_csb 3049   <.cop 3586   ` cfv 5198   1stc1st 6117   2ndc2nd 6118

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by:  dfmpo  6202  f1od2  6214