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Theorem csbhypf 3230
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2945 for class substitution version. (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
csbhypf.1  |-  F/_ x A
csbhypf.2  |-  F/_ x C
csbhypf.3  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
csbhypf  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem csbhypf
StepHypRef Expression
1 csbhypf.1 . . . 4  |-  F/_ x A
21nfeq2 2535 . . 3  |-  F/ x  y  =  A
3 nfcsb1v 3227 . . . 4  |-  F/_ x [_ y  /  x ]_ B
4 csbhypf.2 . . . 4  |-  F/_ x C
53, 4nfeq 2531 . . 3  |-  F/ x [_ y  /  x ]_ B  =  C
62, 5nfim 1822 . 2  |-  F/ x
( y  =  A  ->  [_ y  /  x ]_ B  =  C
)
7 eqeq1 2394 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
8 csbeq1a 3203 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
98eqeq1d 2396 . . 3  |-  ( x  =  y  ->  ( B  =  C  <->  [_ y  /  x ]_ B  =  C ) )
107, 9imbi12d 312 . 2  |-  ( x  =  y  ->  (
( x  =  A  ->  B  =  C )  <->  ( y  =  A  ->  [_ y  /  x ]_ B  =  C ) ) )
11 csbhypf.3 . 2  |-  ( x  =  A  ->  B  =  C )
126, 10, 11chvar 2023 1  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   F/_wnfc 2511   [_csb 3195
This theorem is referenced by:  disji2  4141  disjprg  4150  disjxun  4152  tfisi  4779  iundisj2  19311  disji2f  23864  disjif2  23868  iundisj2f  23874  iundisj2fi  23992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-sbc 3106  df-csb 3196
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