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Theorem pm13.194 27522
Description: Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.194  |-  ( (
ph  /\  x  =  y )  <->  ( [
y  /  x ] ph  /\  ph  /\  x  =  y ) )

Proof of Theorem pm13.194
StepHypRef Expression
1 pm13.13a 27517 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  [. y  /  x ]. ph )
2 sbsbc 3152 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylibr 204 . . 3  |-  ( (
ph  /\  x  =  y )  ->  [ y  /  x ] ph )
4 simpl 444 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ph )
5 simpr 448 . . 3  |-  ( (
ph  /\  x  =  y )  ->  x  =  y )
63, 4, 53jca 1134 . 2  |-  ( (
ph  /\  x  =  y )  ->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y )
)
7 3simpc 956 . 2  |-  ( ( [ y  /  x ] ph  /\  ph  /\  x  =  y )  ->  ( ph  /\  x  =  y ) )
86, 7impbii 181 1  |-  ( (
ph  /\  x  =  y )  <->  ( [
y  /  x ] ph  /\  ph  /\  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936   [wsb 1658   [.wsbc 3148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-sbc 3149
  Copyright terms: Public domain W3C validator