Theorem ifbieq12i 3420

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 3400	. . 3 |- ( A = C -> if ( ph , A , B ) = if ( ph , C , B ) )
3	1, 2	ax-mp 7	. 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 3418	. 2 |- if ( ph , C , B ) = if ( ps , C , D )
7	3, 6	eqtri 2109	1 |- if ( ph , A , B ) = if ( ps , C , D )

Syntax hints:   <-> wb 104   = wceq 1290   ifcif 3397

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-if 3398

This theorem is referenced by: (None)