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Theorem csbhypf 3116
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2833 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 2430 . . 3  |-  F/ x  y  =  A
3 nfcsb1v 3113 . . . 4  |-  F/_ x [_ y  /  x ]_ B
4 csbhypf.2 . . . 4  |-  F/_ x C
53, 4nfeq 2426 . . 3  |-  F/ x [_ y  /  x ]_ B  =  C
62, 5nfim 1769 . 2  |-  F/ x
( y  =  A  ->  [_ y  /  x ]_ B  =  C
)
7 eqeq1 2289 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
8 csbeq1a 3089 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
98eqeq1d 2291 . . 3  |-  ( x  =  y  ->  ( B  =  C  <->  [_ y  /  x ]_ B  =  C ) )
107, 9imbi12d 311 . 2  |-  ( x  =  y  ->  (
( x  =  A  ->  B  =  C )  <->  ( y  =  A  ->  [_ y  /  x ]_ B  =  C ) ) )
11 csbhypf.3 . 2  |-  ( x  =  A  ->  B  =  C )
126, 10, 11chvar 1926 1  |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   F/_wnfc 2406   [_csb 3081
This theorem is referenced by:  disji2  4010  disjprg  4019  disjxun  4021  tfisi  4649  iundisj2  18906  disji2f  23354  disjif2  23358  iundisj2fi  23364  iundisj2f  23366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-sbc 2992  df-csb 3082
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