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Theorem sbcbidv2 24969
Description: Formula-building deduction rule for class substitution with different classes. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
Hypotheses
Ref Expression
sbcbidv2.1  |-  ( ph  ->  A  =  B )
sbcbidv2.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
sbcbidv2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ch ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)    B( x)

Proof of Theorem sbcbidv2
StepHypRef Expression
1 sbcbidv2.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
21sbcbidv 3045 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
3 sbcbidv2.1 . . 3  |-  ( ph  ->  A  =  B )
4 dfsbcq 2993 . . 3  |-  ( A  =  B  ->  ( [. A  /  x ]. ch  <->  [. B  /  x ]. ch ) )
53, 4syl 15 . 2  |-  ( ph  ->  ( [. A  /  x ]. ch  <->  [. B  /  x ]. ch ) )
62, 5bitrd 244 1  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. B  /  x ]. ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   [.wsbc 2991
This theorem is referenced by:  isibg2  26110  isibcg  26191
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-sbc 2992
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