Theorem ifeq12da 4118

Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)

Hypotheses

Ref	Expression
ifeq12da.1	|- ( ( ph /\ ps ) -> A = C )
ifeq12da.2	|- ( ( ph /\ -. ps ) -> B = D )

Assertion

Ref	Expression
ifeq12da	|- ( ph -> if ( ps , A , B ) = if ( ps , C , D ) )

Proof of Theorem ifeq12da

Step	Hyp	Ref	Expression
1		ifeq12da.1	. . . 4 |- ( ( ph /\ ps ) -> A = C )
2	1	ifeq1da 4116	. . 3 |- ( ph -> if ( ps , A , B ) = if ( ps , C , B ) )
3		iftrue 4092	. . . 4 |- ( ps -> if ( ps , C , B ) = C )
4		iftrue 4092	. . . 4 |- ( ps -> if ( ps , C , D ) = C )
5	3, 4	eqtr4d 2659	. . 3 |- ( ps -> if ( ps , C , B ) = if ( ps , C , D ) )
6	2, 5	sylan9eq 2676	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = if ( ps , C , D ) )
7		ifeq12da.2	. . . 4 |- ( ( ph /\ -. ps ) -> B = D )
8	7	ifeq2da 4117	. . 3 |- ( ph -> if ( ps , A , B ) = if ( ps , A , D ) )
9		iffalse 4095	. . . 4 |- ( -. ps -> if ( ps , A , D ) = D )
10		iffalse 4095	. . . 4 |- ( -. ps -> if ( ps , C , D ) = D )
11	9, 10	eqtr4d 2659	. . 3 |- ( -. ps -> if ( ps , A , D ) = if ( ps , C , D ) )
12	8, 11	sylan9eq 2676	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = if ( ps , C , D ) )
13	6, 12	pm2.61dan 832	1 |- ( ph -> if ( ps , A , B ) = if ( ps , C , D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   ifcif 4086

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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087

This theorem is referenced by:  ifbieq12d2  4119