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Theorem sbcopeq1a 5957
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2849 that avoids the existential quantifiers of copsexg 4071). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
sbcopeq1a  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )

Proof of Theorem sbcopeq1a
StepHypRef Expression
1 vex 2622 . . . . 5  |-  x  e. 
_V
2 vex 2622 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 5920 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2093 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 sbceq1a 2849 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ph  <->  [. ( 2nd `  A
)  /  y ]. ph ) )
64, 5syl 14 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  [. ( 2nd `  A )  /  y ]. ph ) )
71, 2op1std 5919 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2093 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 sbceq1a 2849 . . 3  |-  ( x  =  ( 1st `  A
)  ->  ( [. ( 2nd `  A )  /  y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
108, 9syl 14 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 2nd `  A )  / 
y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
116, 10bitr2d 187 1  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   [.wsbc 2840   <.cop 3449   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  dfopab2  5959  dfoprab3s  5960
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