Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifpimim Structured version   Unicode version

Theorem ifpimim 35852
Description: Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
Assertion
Ref Expression
ifpimim  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
) )

Proof of Theorem ifpimim
StepHypRef Expression
1 pm2.521 157 . . . . . 6  |-  ( -.  ( -.  ph  ->  ph )  ->  ( ph  ->  -.  ph ) )
21orim1i 519 . . . . 5  |-  ( ( -.  ( -.  ph  ->  ph )  \/  ( ps  ->  ch ) )  ->  ( ( ph  ->  -.  ph )  \/  ( ps  ->  ch ) ) )
32adantr 466 . . . 4  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( ( ph  ->  -.  ph )  \/  ( ps  ->  ch ) ) )
4 id 23 . . . . . 6  |-  ( ph  ->  ph )
54orci 391 . . . . 5  |-  ( (
ph  ->  ph )  \/  ( th  ->  ch ) )
65a1i 11 . . . 4  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( ( ph  ->  ph )  \/  ( th  ->  ch ) ) )
73, 6jca 534 . . 3  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( (
( ph  ->  -.  ph )  \/  ( ps  ->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th 
->  ch ) ) ) )
84orci 391 . . . . 5  |-  ( (
ph  ->  ph )  \/  ( ps  ->  ta ) )
98a1i 11 . . . 4  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( ( ph  ->  ph )  \/  ( ps  ->  ta ) ) )
10 simpr 462 . . . 4  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( ( -.  ph  ->  ph )  \/  ( th  ->  ta ) ) )
119, 10jca 534 . . 3  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( (
( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  (
( -.  ph  ->  ph )  \/  ( th 
->  ta ) ) ) )
127, 11jca 534 . 2  |-  ( ( ( -.  ( -. 
ph  ->  ph )  \/  ( ps  ->  ch ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) )  ->  ( (
( ( ph  ->  -. 
ph )  \/  ( ps  ->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th  ->  ch ) ) )  /\  ( ( ( ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  (
( -.  ph  ->  ph )  \/  ( th 
->  ta ) ) ) ) )
13 pm4.81 367 . . . . 5  |-  ( ( -.  ph  ->  ph )  <->  ph )
1413bicomi 205 . . . 4  |-  ( ph  <->  ( -.  ph  ->  ph )
)
15 ifpbi1 35820 . . . 4  |-  ( (
ph 
<->  ( -.  ph  ->  ph ) )  ->  (if- ( ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <-> if- ( ( -.  ph  ->  ph ) ,  ( ps  ->  ch ) ,  ( th  ->  ta ) ) ) )
1614, 15ax-mp 5 . . 3  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <-> if- ( ( -.  ph  ->  ph ) ,  ( ps  ->  ch ) ,  ( th  ->  ta ) ) )
17 dfifp4 1424 . . 3  |-  (if- ( ( -.  ph  ->  ph ) ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  ( ( -.  ( -.  ph  ->  ph )  \/  ( ps 
->  ch ) )  /\  ( ( -.  ph  ->  ph )  \/  ( th  ->  ta ) ) ) )
1816, 17bitri 252 . 2  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  <->  ( ( -.  ( -.  ph  ->  ph )  \/  ( ps 
->  ch ) )  /\  ( ( -.  ph  ->  ph )  \/  ( th  ->  ta ) ) ) )
19 ifpim123g 35843 . 2  |-  ( (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
)  <->  ( ( ( ( ph  ->  -.  ph )  \/  ( ps 
->  ch ) )  /\  ( ( ph  ->  ph )  \/  ( th 
->  ch ) ) )  /\  ( ( (
ph  ->  ph )  \/  ( ps  ->  ta ) )  /\  ( ( -. 
ph  ->  ph )  \/  ( th  ->  ta ) ) ) ) )
2012, 18, 193imtr4i 269 1  |-  (if- (
ph ,  ( ps 
->  ch ) ,  ( th  ->  ta )
)  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370  if-wif 1420
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 188  df-or 371  df-an 372  df-ifp 1421
This theorem is referenced by:  frege58acor  36109  frege60a  36111  frege65a  36116
  Copyright terms: Public domain W3C validator