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Theorem sbcbidv2 25072
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 3058 . 2  |-  ( ph  ->  ( [. A  /  x ]. ps  <->  [. A  /  x ]. ch ) )
3 sbcbidv2.1 . . 3  |-  ( ph  ->  A  =  B )
4 dfsbcq 3006 . . 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 1632   [.wsbc 3004
This theorem is referenced by:  isibg2  26213  isibcg  26294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-sbc 3005
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