Theorem ifbieq2i 3594

Description: Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
ifbieq2i.1	|- ( ph <-> ps )
ifbieq2i.2	|- A = B

Assertion

Ref	Expression
ifbieq2i	|- if ( ph , C , A ) = if ( ps , C , B )

Proof of Theorem ifbieq2i

Step	Hyp	Ref	Expression
1		ifbieq2i.1	. . 3 |- ( ph <-> ps )
2		ifbi 3591	. . 3 |- ( ( ph <-> ps ) -> if ( ph , C , A ) = if ( ps , C , A ) )
3	1, 2	ax-mp 5	. 2 |- if ( ph , C , A ) = if ( ps , C , A )
4		ifbieq2i.2	. . 3 |- A = B
5		ifeq2 3575	. . 3 |- ( A = B -> if ( ps , C , A ) = if ( ps , C , B ) )
6	4, 5	ax-mp 5	. 2 |- if ( ps , C , A ) = if ( ps , C , B )
7	3, 6	eqtri 2226	1 |- if ( ph , C , A ) = if ( ps , C , B )

Syntax hints:   <-> wb 105   = wceq 1373   ifcif 3571

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-if 3572

This theorem is referenced by:  ifbieq12i  3596  gcdcom  12294  gcdass  12336  lcmcom  12386  lcmass  12407