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Theorem cdleme31fv2 29271
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
cdleme31fv2.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
Assertion
Ref Expression
cdleme31fv2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Distinct variable groups:    x, B    x, 
.<_    x, P    x, Q    x, W    x, X
Allowed substitution hints:    F( x)    O( x)

Proof of Theorem cdleme31fv2
StepHypRef Expression
1 cdleme31fv2.f . . 3  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )
21a1i 12 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) ) )
3 breq1 3923 . . . . . . . . 9  |-  ( x  =  X  ->  (
x  .<_  W  <->  X  .<_  W ) )
43notbid 287 . . . . . . . 8  |-  ( x  =  X  ->  ( -.  x  .<_  W  <->  -.  X  .<_  W ) )
54anbi2d 687 . . . . . . 7  |-  ( x  =  X  ->  (
( P  =/=  Q  /\  -.  x  .<_  W )  <-> 
( P  =/=  Q  /\  -.  X  .<_  W ) ) )
65notbid 287 . . . . . 6  |-  ( x  =  X  ->  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  <->  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) )
76biimparc 475 . . . . 5  |-  ( ( -.  ( P  =/= 
Q  /\  -.  X  .<_  W )  /\  x  =  X )  ->  -.  ( P  =/=  Q  /\  -.  x  .<_  W ) )
87adantll 697 . . . 4  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  -.  ( P  =/= 
Q  /\  -.  x  .<_  W ) )
9 iffalse 3477 . . . 4  |-  ( -.  ( P  =/=  Q  /\  -.  x  .<_  W )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x
)  =  x )
108, 9syl 17 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  x )
11 simpr 449 . . 3  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  x  =  X )
1210, 11eqtrd 2285 . 2  |-  ( ( ( X  e.  B  /\  -.  ( P  =/= 
Q  /\  -.  X  .<_  W ) )  /\  x  =  X )  ->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x )  =  X )
13 simpl 445 . 2  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  X  e.  B )
142, 12, 13, 13fvmptd 5458 1  |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  X )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   ifcif 3470   class class class wbr 3920    e. cmpt 3974   ` cfv 4592
This theorem is referenced by:  cdleme31id  29272  cdleme32fvcl  29318  cdleme32e  29323  cdleme32le  29325  cdleme48gfv  29415  cdleme50ldil  29426
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-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608
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