Theorem elimdelov 6736

Description: Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (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

Step	Hyp	Ref	Expression
1		elimdelov.1	. . 3 |- ( ph -> C e. ( A F B ) )
2		iftrue 4092	. . 3 |- ( ph -> if ( ph , C , Z ) = C )
3		iftrue 4092	. . . 4 |- ( ph -> if ( ph , A , X ) = A )
4		iftrue 4092	. . . 4 |- ( ph -> if ( ph , B , Y ) = B )
5	3, 4	oveq12d 6668	. . 3 |- ( ph -> ( if ( ph , A , X ) F if ( ph , B , Y ) ) = ( A F B ) )
6	1, 2, 5	3eltr4d 2716	. 2 |- ( ph -> if ( ph , C , Z ) e. ( if ( ph , A , X ) F if ( ph , B , Y ) ) )
7		iffalse 4095	. . . 4 |- ( -. ph -> if ( ph , C , Z ) = Z )
8		elimdelov.2	. . . 4 |- Z e. ( X F Y )
9	7, 8	syl6eqel 2709	. . 3 |- ( -. ph -> if ( ph , C , Z ) e. ( X F Y ) )
10		iffalse 4095	. . . 4 |- ( -. ph -> if ( ph , A , X ) = X )
11		iffalse 4095	. . . 4 |- ( -. ph -> if ( ph , B , Y ) = Y )
12	10, 11	oveq12d 6668	. . 3 |- ( -. ph -> ( if ( ph , A , X ) F if ( ph , B , Y ) ) = ( X F Y ) )
13	9, 12	eleqtrrd 2704	. 2 |- ( -. ph -> if ( ph , C , Z ) e. ( if ( ph , A , X ) F if ( ph , B , Y ) ) )
14	6, 13	pm2.61i 176	1 |- if ( ph , C , Z ) e. ( if ( ph , A , X ) F if ( ph , B , Y ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   ifcif 4086  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)