Theorem ifbieq2i 3603

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 3600	. . 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 3583	. . 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 2228	1 |- if ( ph , C , A ) = if ( ps , C , B )

Syntax hints:   <-> wb 105   = wceq 1373   ifcif 3579

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-un 3178  df-if 3580

This theorem is referenced by:  ifbieq12i  3605  gcdcom  12409  gcdass  12451  lcmcom  12501  lcmass  12522