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Theorem csbopeq1a 6401
Description: Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 3260). (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
StepHypRef Expression
1 vex 2960 . . . . 5  |-  x  e. 
_V
2 vex 2960 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 6359 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2442 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 csbeq1a 3260 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  B  =  [_ ( 2nd `  A
)  /  y ]_ B )
64, 5syl 16 . 2  |-  ( A  =  <. x ,  y
>.  ->  B  =  [_ ( 2nd `  A )  /  y ]_ B
)
71, 2op1std 6358 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2442 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 csbeq1a 3260 . . 3  |-  ( x  =  ( 1st `  A
)  ->  [_ ( 2nd `  A )  /  y ]_ B  =  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A )  /  y ]_ B )
108, 9syl 16 . 2  |-  ( A  =  <. x ,  y
>.  ->  [_ ( 2nd `  A
)  /  y ]_ B  =  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A
)  /  y ]_ B )
116, 10eqtr2d 2470 1  |-  ( A  =  <. x ,  y
>.  ->  [_ ( 1st `  A
)  /  x ]_ [_ ( 2nd `  A
)  /  y ]_ B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   [_csb 3252   <.cop 3818   ` cfv 5455   1stc1st 6348   2ndc2nd 6349
This theorem is referenced by:  dfmpt2  6438  wdom2d2  27107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fv 5463  df-1st 6350  df-2nd 6351
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