MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elimdelov Structured version   Unicode version

Theorem elimdelov 6351
Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 29296 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 3935 . . 3  |-  ( ph  ->  if ( ph ,  C ,  Z )  =  C )
3 iftrue 3935 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  X )  =  A )
4 iftrue 3935 . . . 4  |-  ( ph  ->  if ( ph ,  B ,  Y )  =  B )
53, 4oveq12d 6288 . . 3  |-  ( ph  ->  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( A F B ) )
61, 2, 53eltr4d 2557 . 2  |-  ( ph  ->  if ( ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
7 iffalse 3938 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  =  Z )
8 elimdelov.2 . . . 4  |-  Z  e.  ( X F Y )
97, 8syl6eqel 2550 . . 3  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( X F Y ) )
10 iffalse 3938 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  A ,  X )  =  X )
11 iffalse 3938 . . . 4  |-  ( -. 
ph  ->  if ( ph ,  B ,  Y )  =  Y )
1210, 11oveq12d 6288 . . 3  |-  ( -. 
ph  ->  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) )  =  ( X F Y ) )
139, 12eleqtrrd 2545 . 2  |-  ( -. 
ph  ->  if ( ph ,  C ,  Z )  e.  ( if (
ph ,  A ,  X ) F if ( ph ,  B ,  Y ) ) )
146, 13pm2.61i 164 1  |-  if (
ph ,  C ,  Z )  e.  ( if ( ph ,  A ,  X ) F if ( ph ,  B ,  Y )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1823   ifcif 3929  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  ghomgrplem  29296
  Copyright terms: Public domain W3C validator