Theorem ifeq2d 3523

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 3509	. 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 1335   ifcif 3505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-un 3106  df-if 3506

This theorem is referenced by:  ifeq12d  3524  ifbieq2d  3529  exp3val  10421  xrmaxiflemcom  11146  peano4nninf  13578  peano3nninf  13579