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Theorem elimdelov 5927
Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 23996 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
Hypotheses
Ref Expression
elimdelov.1  |-  ( ph  ->  C  e.  ( A F B ) )
elimdelov.2  |-  Z  e.  ( X F Y )
Assertion
Ref Expression
elimdelov  |-  if (
ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y )
)

Proof of Theorem elimdelov
StepHypRef Expression
1 iftrue 3571 . . . 4  |-  ( ph  ->  if ( ph ,  C ,  Z )  =  C )
2 elimdelov.1 . . . 4  |-  ( ph  ->  C  e.  ( A F B ) )
31, 2eqeltrd 2357 . . 3  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( A F B ) )
4 iftrue 3571 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  X )  =  A )
5 iftrue 3571 . . . 4  |-  ( ph  ->  if ( ph ,  B ,  Y )  =  B )
64, 5oveq12d 5876 . . 3  |-  ( ph  ->  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( A F B ) )
73, 6eleqtrrd 2360 . 2  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
8 iffalse 3572 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  =  Z )
9 elimdelov.2 . . . 4  |-  Z  e.  ( X F Y )
108, 9syl6eqel 2371 . . 3  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( X F Y ) )
11 iffalse 3572 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  X )  =  X )
12 iffalse 3572 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  B ,  Y )  =  Y )
1311, 12oveq12d 5876 . . 3  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( X F Y ) )
1410, 13eleqtrrd 2360 . 2  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
157, 14pm2.61i 156 1  |-  if (
ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1684   ifcif 3565  (class class class)co 5858
This theorem is referenced by:  ghomgrplem  23996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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