Theorem ifeq2d 4105

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 4091	. 2 |- ( A = B -> if ( ps , C , A ) = if ( ps , C , B ) )
3	1, 2	syl 17	1 |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )

Syntax hints:   -> wi 4   = wceq 1483   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-un 3579  df-if 4087

This theorem is referenced by:  ifeq12d  4106  ifbieq2d  4111  ifeq2da  4117  ifcomnan  4137  rdgeq1  7507  cantnflem1d  8585  cantnflem1  8586  rexmul  12101  1arithlem4  15630  ramcl  15733  mplcoe1  19465  mplcoe5  19468  subrgascl  19498  scmatscm  20319  marrepfval  20366  ma1repveval  20377  mulmarep1el  20378  mdetralt2  20415  mdetunilem8  20425  maduval  20444  maducoeval2  20446  madurid  20450  minmar1val0  20453  monmatcollpw  20584  pmatcollpwscmatlem1  20594  monmat2matmon  20629  itg2monolem1  23517  iblmulc2  23597  itgmulc2lem1  23598  bddmulibl  23605  dvtaylp  24124  dchrinvcl  24978  rpvmasum2  25201  padicfval  25305  plymulx  30625  itg2addnclem  33461  itg2addnclem3  33463  itg2addnc  33464  itgmulc2nclem1  33476  hdmap1fval  37086  itgioocnicc  40193  etransclem14  40465  etransclem17  40468  etransclem21  40472  etransclem25  40476  etransclem28  40479  etransclem31  40482  hsphoif  40790  hoidmvval  40791  hsphoival  40793  hoidmvlelem5  40813  hoidmvle  40814  ovnhoi  40817  hspmbllem2  40841