Theorem ifeq2d 3413

Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)

Hypothesis

Ref	Expression
ifeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
ifeq2d	|- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Proof of Theorem ifeq2d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 |- ( ph -> A = B )
2		ifeq2 3401	. 2 |- ( A = B -> if ( ps , C , A ) = if ( ps , C , B ) )
3	1, 2	syl 14	1 |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Syntax hints:   -> wi 4   = wceq 1290   ifcif 3397

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rab 2369  df-v 2622  df-un 3004  df-if 3398

This theorem is referenced by:  ifeq12d  3414  ifbieq2d  3419  exp3val  10011  peano4nninf  12162  peano3nninf  12163