Theorem ifbieq12d2 4119

Description: Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)

Hypotheses

Ref	Expression
ifbieq12d2.1	|- ( ph -> ( ps <-> ch ) )
ifbieq12d2.2	|- ( ( ph /\ ps ) -> A = C )
ifbieq12d2.3	|- ( ( ph /\ -. ps ) -> B = D )

Assertion

Ref	Expression
ifbieq12d2	|- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Proof of Theorem ifbieq12d2

Step	Hyp	Ref	Expression
1		ifbieq12d2.2	. . 3 |- ( ( ph /\ ps ) -> A = C )
2		ifbieq12d2.3	. . 3 |- ( ( ph /\ -. ps ) -> B = D )
3	1, 2	ifeq12da 4118	. 2 |- ( ph -> if ( ps , A , B ) = if ( ps , C , D ) )
4		ifbieq12d2.1	. . 3 |- ( ph -> ( ps <-> ch ) )
5	4	ifbid 4108	. 2 |- ( ph -> if ( ps , C , D ) = if ( ch , C , D ) )
6	3, 5	eqtrd 2656	1 |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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:  ofccat  13708  itgeq12dv  30388  sgnneg  30602