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Theorem bj-frege52a 37269
Description: PLEASE DESCRIBE ME.

Part of Axiom 52 of [Frege1879] p. 50. (Contributed by Richard Penner, 24-Dec-2019.)

Assertion
Ref Expression
bj-frege52a  |-  ( (
ph 
<->  ps )  ->  (if- ( ph ,  th ,  ch )  -> if- ( ps ,  th ,  ch ) ) )

Proof of Theorem bj-frege52a
StepHypRef Expression
1 df-bj-if 33626 . . 3  |-  (if- (
ph ,  th ,  ch )  <->  ( ( ph  ->  th )  /\  ( -.  ph  ->  ch )
) )
21biimpi 194 . 2  |-  (if- (
ph ,  th ,  ch )  ->  ( (
ph  ->  th )  /\  ( -.  ph  ->  ch )
) )
3 bi2 198 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
43imim1d 75 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( ph  ->  th )  ->  ( ps  ->  th )
) )
5 bi1 186 . . . . 5  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
6 con3 134 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( -.  ps  ->  -. 
ph ) )
75, 6syl 16 . . . 4  |-  ( (
ph 
<->  ps )  ->  ( -.  ps  ->  -.  ph )
)
87imim1d 75 . . 3  |-  ( (
ph 
<->  ps )  ->  (
( -.  ph  ->  ch )  ->  ( -.  ps  ->  ch ) ) )
94, 8anim12d 563 . 2  |-  ( (
ph 
<->  ps )  ->  (
( ( ph  ->  th )  /\  ( -. 
ph  ->  ch ) )  ->  ( ( ps 
->  th )  /\  ( -.  ps  ->  ch )
) ) )
10 df-bj-if 33626 . . 3  |-  (if- ( ps ,  th ,  ch )  <->  ( ( ps 
->  th )  /\  ( -.  ps  ->  ch )
) )
1110biimpri 206 . 2  |-  ( ( ( ps  ->  th )  /\  ( -.  ps  ->  ch ) )  -> if- ( ps ,  th ,  ch ) )
122, 9, 11syl56 34 1  |-  ( (
ph 
<->  ps )  ->  (if- ( ph ,  th ,  ch )  -> if- ( ps ,  th ,  ch ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369  if-wif 33625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-an 371  df-bj-if 33626
This theorem is referenced by:  bj-frege53a  37271  bj-frege57a  37283
  Copyright terms: Public domain W3C validator