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Theorem csbhypf 3454
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3160 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 2646 . . 3  |-  F/ x  y  =  A
3 nfcsb1v 3451 . . . 4  |-  F/_ x [_ y  /  x ]_ B
4 csbhypf.2 . . . 4  |-  F/_ x C
53, 4nfeq 2640 . . 3  |-  F/ x [_ y  /  x ]_ B  =  C
62, 5nfim 1867 . 2  |-  F/ x
( y  =  A  ->  [_ y  /  x ]_ B  =  C
)
7 eqeq1 2471 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
8 csbeq1a 3444 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
98eqeq1d 2469 . . 3  |-  ( x  =  y  ->  ( B  =  C  <->  [_ y  /  x ]_ B  =  C ) )
107, 9imbi12d 320 . 2  |-  ( x  =  y  ->  (
( x  =  A  ->  B  =  C )  <->  ( y  =  A  ->  [_ y  /  x ]_ B  =  C ) ) )
11 csbhypf.3 . 2  |-  ( x  =  A  ->  B  =  C )
126, 10, 11chvar 1982 1  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   F/_wnfc 2615   [_csb 3435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-sbc 3332  df-csb 3436
This theorem is referenced by:  disji2  4434  disjprg  4443  disjxun  4445  tfisi  6678  coe1fzgsumdlem  18154  evl1gsumdlem  18203  iundisj2  21786  disji2f  27208  disjif2  27212  iundisj2f  27219  iundisj2fi  27367
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