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Theorem pm13.194 26779
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 26774 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  [. y  /  x ]. ph )
2 sbsbc 2925 . . . 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 1137 . 2  |-  ( (
ph  /\  x  =  y )  ->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y )
)
7 3simpc 959 . 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 939    = wceq 1619   [wsb 1882   [.wsbc 2921
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-17 1628  ax-12o 1664  ax-9 1684  ax-4 1692  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 941  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-sbc 2922
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