Theorem ovif12 6739

Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
ovif12	|- ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) )

Proof of Theorem ovif12

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . 4 |- ( ph -> if ( ph , A , B ) = A )
2		iftrue 4092	. . . 4 |- ( ph -> if ( ph , C , D ) = C )
3	1, 2	oveq12d 6668	. . 3 |- ( ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = ( A F C ) )
4		iftrue 4092	. . 3 |- ( ph -> if ( ph , ( A F C ) , ( B F D ) ) = ( A F C ) )
5	3, 4	eqtr4d 2659	. 2 |- ( ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) ) )
6		iffalse 4095	. . . 4 |- ( -. ph -> if ( ph , A , B ) = B )
7		iffalse 4095	. . . 4 |- ( -. ph -> if ( ph , C , D ) = D )
8	6, 7	oveq12d 6668	. . 3 |- ( -. ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = ( B F D ) )
9		iffalse 4095	. . 3 |- ( -. ph -> if ( ph , ( A F C ) , ( B F D ) ) = ( B F D ) )
10	8, 9	eqtr4d 2659	. 2 |- ( -. ph -> ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) ) )
11	5, 10	pm2.61i 176	1 |- ( if ( ph , A , B ) F if ( ph , C , D ) ) = if ( ph , ( A F C ) , ( B F D ) )

Syntax hints:   -. wn 3   = wceq 1483   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:  ofccat  13708  limccnp2  23656  ftc1anclem5  33489