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Theorem csbhypf 3008
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2709 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 2270 . . 3  |-  F/ x  y  =  A
3 nfcsb1v 3005 . . . 4  |-  F/_ x [_ y  /  x ]_ B
4 csbhypf.2 . . . 4  |-  F/_ x C
53, 4nfeq 2266 . . 3  |-  F/ x [_ y  /  x ]_ B  =  C
62, 5nfim 1536 . 2  |-  F/ x
( y  =  A  ->  [_ y  /  x ]_ B  =  C
)
7 eqeq1 2124 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
8 csbeq1a 2983 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
98eqeq1d 2126 . . 3  |-  ( x  =  y  ->  ( B  =  C  <->  [_ y  /  x ]_ B  =  C ) )
107, 9imbi12d 233 . 2  |-  ( x  =  y  ->  (
( x  =  A  ->  B  =  C )  <->  ( y  =  A  ->  [_ y  /  x ]_ B  =  C ) ) )
11 csbhypf.3 . 2  |-  ( x  =  A  ->  B  =  C )
126, 10, 11chvar 1715 1  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   F/_wnfc 2245   [_csb 2975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-sbc 2883  df-csb 2976
This theorem is referenced by:  disji2  3892  tfisi  4471
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