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Theorem elimdelov 5943
Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 24011 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 3584 . . . 4  |-  ( ph  ->  if ( ph ,  C ,  Z )  =  C )
2 elimdelov.1 . . . 4  |-  ( ph  ->  C  e.  ( A F B ) )
31, 2eqeltrd 2370 . . 3  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( A F B ) )
4 iftrue 3584 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  X )  =  A )
5 iftrue 3584 . . . 4  |-  ( ph  ->  if ( ph ,  B ,  Y )  =  B )
64, 5oveq12d 5892 . . 3  |-  ( ph  ->  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( A F B ) )
73, 6eleqtrrd 2373 . 2  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
8 iffalse 3585 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  =  Z )
9 elimdelov.2 . . . 4  |-  Z  e.  ( X F Y )
108, 9syl6eqel 2384 . . 3  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( X F Y ) )
11 iffalse 3585 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  X )  =  X )
12 iffalse 3585 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  B ,  Y )  =  Y )
1311, 12oveq12d 5892 . . 3  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( X F Y ) )
1410, 13eleqtrrd 2373 . 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 1696   ifcif 3578  (class class class)co 5874
This theorem is referenced by:  ghomgrplem  24011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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