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Theorem sbcopeq1a 6399
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3171 that avoids the existential quantifiers of copsexg 4442). (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 2959 . . . . 5  |-  x  e. 
_V
2 vex 2959 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 6358 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2441 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 sbceq1a 3171 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ph  <->  [. ( 2nd `  A
)  /  y ]. ph ) )
64, 5syl 16 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  [. ( 2nd `  A )  /  y ]. ph ) )
71, 2op1std 6357 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2441 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 sbceq1a 3171 . . 3  |-  ( x  =  ( 1st `  A
)  ->  ( [. ( 2nd `  A )  /  y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
108, 9syl 16 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 2nd `  A )  / 
y ]. ph  <->  [. ( 1st `  A )  /  x ]. [. ( 2nd `  A
)  /  y ]. ph ) )
116, 10bitr2d 246 1  |-  ( A  =  <. x ,  y
>.  ->  ( [. ( 1st `  A )  /  x ]. [. ( 2nd `  A )  /  y ]. ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   [.wsbc 3161   <.cop 3817   ` cfv 5454   1stc1st 6347   2ndc2nd 6348
This theorem is referenced by:  dfopab2  6401  dfoprab3s  6402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-1st 6349  df-2nd 6350
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