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Theorem csbopeq1a 6189
Description: Equality theorem for substitution of a class  A for an ordered pair  <. x ,  y >. in  B (analog of csbeq1a 3102). (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 2804 . . . . 5  |-  x  e. 
_V
2 vex 2804 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 6147 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2301 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 csbeq1a 3102 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  B  =  [_ ( 2nd `  A
)  /  y ]_ B )
64, 5syl 15 . 2  |-  ( A  =  <. x ,  y
>.  ->  B  =  [_ ( 2nd `  A )  /  y ]_ B
)
71, 2op1std 6146 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2301 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 csbeq1a 3102 . . 3  |-  ( x  =  ( 1st `  A
)  ->  [_ ( 2nd `  A )  /  y ]_ B  =  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A )  /  y ]_ B )
108, 9syl 15 . 2  |-  ( A  =  <. x ,  y
>.  ->  [_ ( 2nd `  A
)  /  y ]_ B  =  [_ ( 1st `  A )  /  x ]_ [_ ( 2nd `  A
)  /  y ]_ B )
116, 10eqtr2d 2329 1  |-  ( A  =  <. x ,  y
>.  ->  [_ ( 1st `  A
)  /  x ]_ [_ ( 2nd `  A
)  /  y ]_ B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   [_csb 3094   <.cop 3656   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dfmpt2  6225  wdom2d2  27231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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