Theorem ifbieq12i 3586

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 3564	. . 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 3584	. 2 |- if ( ph , C , B ) = if ( ps , C , D )
7	3, 6	eqtri 2217	1 |- if ( ph , A , B ) = if ( ps , C , D )

Syntax hints:   <-> wb 105   = wceq 1364   ifcif 3561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-un 3161  df-if 3562

This theorem is referenced by: (None)