Theorem ifbieq12i 4112

Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Hypotheses

Ref	Expression
ifbieq12i.1	|- ( ph <-> ps )
ifbieq12i.2	|- A = C
ifbieq12i.3	|- B = D

Assertion

Ref	Expression
ifbieq12i	|- if ( ph , A , B ) = if ( ps , C , D )

Proof of Theorem ifbieq12i

Step	Hyp	Ref	Expression
1		ifbieq12i.2	. . 3 |- A = C
2		ifeq1 4090	. . 3 |- ( A = C -> if ( ph , A , B ) = if ( ph , C , B ) )
3	1, 2	ax-mp 5	. 2 |- if ( ph , A , B ) = if ( ph , C , B )
4		ifbieq12i.1	. . 3 |- ( ph <-> ps )
5		ifbieq12i.3	. . 3 |- B = D
6	4, 5	ifbieq2i 4110	. 2 |- if ( ph , C , B ) = if ( ps , C , D )
7	3, 6	eqtri 2644	1 |- if ( ph , A , B ) = if ( ps , C , D )

Syntax hints:   <-> wb 196   = 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:  cbvditg  23618  sgnneg  30602  binomcxplemdvsum  38554