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Theorem pm13.194 27627
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 27622 . . . 4  |-  ( (
ph  /\  x  =  y )  ->  [. y  /  x ]. ph )
2 sbsbc 3171 . . . 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 1135 . 2  |-  ( (
ph  /\  x  =  y )  ->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y )
)
7 3simpc 957 . 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 937   [wsb 1659   [.wsbc 3167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-11 1763  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-sbc 3168
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