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Theorem ifnefals 3603
Description: Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
Assertion
Ref Expression
ifnefals  |-  ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  ->  -.  ph )

Proof of Theorem ifnefals
StepHypRef Expression
1 iftrue 3566 . . 3  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21adantl 277 . 2  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  if ( ph ,  A ,  B )  =  A )
3 simplr 528 . . . 4  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  if ( ph ,  A ,  B )  =  B )
4 simpll 527 . . . . 5  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  A  =/=  B
)
54necomd 2453 . . . 4  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  B  =/=  A
)
63, 5eqnetrd 2391 . . 3  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  if ( ph ,  A ,  B )  =/=  A )
76neneqd 2388 . 2  |-  ( ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  /\  ph )  ->  -.  if ( ph ,  A ,  B )  =  A )
82, 7pm2.65da 662 1  |-  ( ( A  =/=  B  /\  if ( ph ,  A ,  B )  =  B )  ->  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    = wceq 1364    =/= wne 2367   ifcif 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ne 2368  df-if 3562
This theorem is referenced by:  ifnebibdc  3604
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