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Theorem sbcopeq1a 6172
Description: Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 3001 that avoids the existential quantifiers of copsexg 4252). (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 2791 . . . . 5  |-  x  e. 
_V
2 vex 2791 . . . . 5  |-  y  e. 
_V
31, 2op2ndd 6131 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  y )
43eqcomd 2288 . . 3  |-  ( A  =  <. x ,  y
>.  ->  y  =  ( 2nd `  A ) )
5 sbceq1a 3001 . . 3  |-  ( y  =  ( 2nd `  A
)  ->  ( ph  <->  [. ( 2nd `  A
)  /  y ]. ph ) )
64, 5syl 15 . 2  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  [. ( 2nd `  A )  /  y ]. ph ) )
71, 2op1std 6130 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  x )
87eqcomd 2288 . . 3  |-  ( A  =  <. x ,  y
>.  ->  x  =  ( 1st `  A ) )
9 sbceq1a 3001 . . 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 1623   [.wsbc 2991   <.cop 3643   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dfopab2  6174  dfoprab3s  6175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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