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Theorem frege55lem1c 38133
Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
frege55lem1c  |-  ( (
ph  ->  [. A  /  x ]. x  =  B
)  ->  ( ph  ->  A  =  B ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem frege55lem1c
StepHypRef Expression
1 df-sbc 3328 . . 3  |-  ( [. A  /  x ]. x  =  B  <->  A  e.  { x  |  x  =  B } )
2 ax-frege1 38003 . . . . 5  |-  ( A  e.  { x  |  x  =  B }  ->  ( A  =  B  ->  A  e.  {
x  |  x  =  B } ) )
3 eqeq1 2461 . . . . . 6  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
43elab3g 3252 . . . . 5  |-  ( ( A  =  B  ->  A  e.  { x  |  x  =  B } )  ->  ( A  e.  { x  |  x  =  B } 
<->  A  =  B ) )
52, 4syl 16 . . . 4  |-  ( A  e.  { x  |  x  =  B }  ->  ( A  e.  {
x  |  x  =  B }  <->  A  =  B ) )
65ibi 241 . . 3  |-  ( A  e.  { x  |  x  =  B }  ->  A  =  B )
71, 6sylbi 195 . 2  |-  ( [. A  /  x ]. x  =  B  ->  A  =  B )
87imim2i 14 1  |-  ( (
ph  ->  [. A  /  x ]. x  =  B
)  ->  ( ph  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   {cab 2442   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-frege1 38003
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328
This theorem is referenced by:  frege56c  38136
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