Theorem ifbieq2i 3580

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 3577	. . 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 3561	. . 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 2214	1 |- if ( ph , C , A ) = if ( ps , C , B )

Syntax hints:   <-> wb 105   = wceq 1364   ifcif 3557

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-un 3157  df-if 3558

This theorem is referenced by:  ifbieq12i  3582  gcdcom  12110  gcdass  12152  lcmcom  12202  lcmass  12223