Theorem ifbieq12i 3551

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 3529	. . 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 3549	. 2 |- if ( ph , C , B ) = if ( ps , C , D )
7	3, 6	eqtri 2191	1 |- if ( ph , A , B ) = if ( ps , C , D )

Syntax hints:   <-> wb 104   = wceq 1348   ifcif 3526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-un 3125  df-if 3527

This theorem is referenced by: (None)