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Theorem elimdelov 6153
Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 25100 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 elimdelov.1 . . 3  |-  ( ph  ->  C  e.  ( A F B ) )
2 iftrue 3745 . . 3  |-  ( ph  ->  if ( ph ,  C ,  Z )  =  C )
3 iftrue 3745 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  X )  =  A )
4 iftrue 3745 . . . 4  |-  ( ph  ->  if ( ph ,  B ,  Y )  =  B )
53, 4oveq12d 6099 . . 3  |-  ( ph  ->  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( A F B ) )
61, 2, 53eltr4d 2517 . 2  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
7 iffalse 3746 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  =  Z )
8 elimdelov.2 . . . 4  |-  Z  e.  ( X F Y )
97, 8syl6eqel 2524 . . 3  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( X F Y ) )
10 iffalse 3746 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  X )  =  X )
11 iffalse 3746 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  B ,  Y )  =  Y )
1210, 11oveq12d 6099 . . 3  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( X F Y ) )
139, 12eleqtrrd 2513 . 2  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
146, 13pm2.61i 158 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 1725   ifcif 3739  (class class class)co 6081
This theorem is referenced by:  ghomgrplem  25100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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