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Theorem sbcopeq1a 6299
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3087 that avoids the existential quantifiers of copsexg 4355). (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 2876 . . . . 5  |-  x  e. 
_V
2 vex 2876 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 6258 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2371 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 sbceq1a 3087 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ph  <->  [. ( 2nd `  A
)  /  y ]. ph ) )
64, 5syl 15 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  [. ( 2nd `  A )  /  y ]. ph ) )
71, 2op1std 6257 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2371 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 sbceq1a 3087 . . 3  |-  ( x  =  ( 1st `  A
)  ->  ( [. ( 2nd `  A )  /  y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
108, 9syl 15 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 2nd `  A )  / 
y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
116, 10bitr2d 245 1  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1647   [.wsbc 3077   <.cop 3732   ` cfv 5358   1stc1st 6247   2ndc2nd 6248
This theorem is referenced by:  dfopab2  6301  dfoprab3s  6302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fv 5366  df-1st 6249  df-2nd 6250
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