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Theorem sbcopeq1a 5844
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2825 that avoids the existential quantifiers of copsexg 4007). (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 2605 . . . . 5  |-  x  e. 
_V
2 vex 2605 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 5807 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2087 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 sbceq1a 2825 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ph  <->  [. ( 2nd `  A
)  /  y ]. ph ) )
64, 5syl 14 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  [. ( 2nd `  A )  /  y ]. ph ) )
71, 2op1std 5806 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2087 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 sbceq1a 2825 . . 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 1285   [.wsbc 2816   <.cop 3409   ` cfv 4932   1stc1st 5796   2ndc2nd 5797
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798  df-2nd 5799
This theorem is referenced by:  dfopab2  5846  dfoprab3s  5847
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