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Theorem frege55lem1c 36557
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 3280 . . 3  |-  ( [. A  /  x ]. x  =  B  <->  A  e.  { x  |  x  =  B } )
2 eqeq1 2466 . . . . 5  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
32elabg 3198 . . . 4  |-  ( A  e.  { x  |  x  =  B }  ->  ( A  e.  {
x  |  x  =  B }  <->  A  =  B ) )
43ibi 249 . . 3  |-  ( A  e.  { x  |  x  =  B }  ->  A  =  B )
51, 4sylbi 200 . 2  |-  ( [. A  /  x ]. x  =  B  ->  A  =  B )
65imim2i 16 1  |-  ( (
ph  ->  [. A  /  x ]. x  =  B
)  ->  ( ph  ->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   {cab 2448   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-sbc 3280
This theorem is referenced by:  frege56c  36560
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