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Theorem csbiedf 3441
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1  |-  F/ x ph
csbiedf.2  |-  ( ph  -> 
F/_ x C )
csbiedf.3  |-  ( ph  ->  A  e.  V )
csbiedf.4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
Assertion
Ref Expression
csbiedf  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    B( x)    C( x)    V( x)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3  |-  F/ x ph
2 csbiedf.4 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  B  =  C )
32ex 432 . . 3  |-  ( ph  ->  ( x  =  A  ->  B  =  C ) )
41, 3alrimi 1882 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  B  =  C ) )
5 csbiedf.3 . . 3  |-  ( ph  ->  A  e.  V )
6 csbiedf.2 . . 3  |-  ( ph  -> 
F/_ x C )
7 csbiebt 3440 . . 3  |-  ( ( A  e.  V  /\  F/_ x C )  -> 
( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
85, 6, 7syl2anc 659 . 2  |-  ( ph  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
)
94, 8mpbid 210 1  |-  ( ph  ->  [_ A  /  x ]_ B  =  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    = wceq 1398   F/wnf 1621    e. wcel 1823   F/_wnfc 2602   [_csb 3420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-sbc 3325  df-csb 3421
This theorem is referenced by:  csbied  3447  csbie2t  3449  natpropd  15464  fucpropd  15465  gsummptf1o  17186  gsummpt2d  28006  sumsnd  31641  fsumsplit1  31812  fprodsplit1f  31832
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