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Theorem pm13.194 27012
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 27007 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  [. y  /  x ]. ph )
2 sbsbc 2997 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylibr 205 . . 3  |-  ( (
ph  /\  x  =  y )  ->  [ y  /  x ] ph )
4 simpl 445 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ph )
5 simpr 449 . . 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 182 1  |-  ( (
ph  /\  x  =  y )  <->  ( [
y  /  x ] ph  /\  ph  /\  x  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1624   [wsb 1631   [.wsbc 2993
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-11 1716  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 938  df-ex 1530  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-sbc 2994
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