Theorem ifbieq2i 3559

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 3556	. . 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 3540	. . 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 2198	1 |- if ( ph , C , A ) = if ( ps , C , B )

Syntax hints:   <-> wb 105   = wceq 1353   ifcif 3536

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-if 3537

This theorem is referenced by:  ifbieq12i  3561  gcdcom  11976  gcdass  12018  lcmcom  12066  lcmass  12087