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Theorem csbhypf 3381
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3094 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 2606 . . 3  |-  F/ x  y  =  A
3 nfcsb1v 3378 . . . 4  |-  F/_ x [_ y  /  x ]_ B
4 csbhypf.2 . . . 4  |-  F/_ x C
53, 4nfeq 2602 . . 3  |-  F/ x [_ y  /  x ]_ B  =  C
62, 5nfim 2002 . 2  |-  F/ x
( y  =  A  ->  [_ y  /  x ]_ B  =  C
)
7 eqeq1 2454 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
8 csbeq1a 3371 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
98eqeq1d 2452 . . 3  |-  ( x  =  y  ->  ( B  =  C  <->  [_ y  /  x ]_ B  =  C ) )
107, 9imbi12d 322 . 2  |-  ( x  =  y  ->  (
( x  =  A  ->  B  =  C )  <->  ( y  =  A  ->  [_ y  /  x ]_ B  =  C ) ) )
11 csbhypf.3 . 2  |-  ( x  =  A  ->  B  =  C )
126, 10, 11chvar 2105 1  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443   F/_wnfc 2578   [_csb 3362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-sbc 3267  df-csb 3363
This theorem is referenced by:  disji2  4388  disjprg  4397  disjxun  4399  tfisi  6682  coe1fzgsumdlem  18888  evl1gsumdlem  18937  iundisj2  22495  disji2f  28180  disjif2  28184  iundisj2f  28193  iundisj2fi  28366
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