Theorem ifbieq2i 3351

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 3348	. . 3 |- ( ( ph <-> ps ) -> if ( ph , C , A ) = if ( ps , C , A ) )
3	1, 2	ax-mp 7	. 2 |- if ( ph , C , A ) = if ( ps , C , A )
4		ifbieq2i.2	. . 3 |- A = B
5		ifeq2 3335	. . 3 |- ( A = B -> if ( ps , C , A ) = if ( ps , C , B ) )
6	4, 5	ax-mp 7	. 2 |- if ( ps , C , A ) = if ( ps , C , B )
7	3, 6	eqtri 2060	1 |- if ( ph , C , A ) = if ( ps , C , B )

Syntax hints:   <-> wb 98   = wceq 1243   ifcif 3331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-if 3332

This theorem is referenced by:  ifbieq12i  3353