Theorem ifeq2d 4055

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

Hypothesis

Ref	Expression
ifeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
ifeq2d	⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2d

Step	Hyp	Ref	Expression
1		ifeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ifeq2 4041	. 2 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:   → wi 4   = wceq 1475  ifcif 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037

This theorem is referenced by:  ifeq12d  4056  ifbieq2d  4061  ifeq2da  4067  ifcomnan  4087  rdgeq1  7394  cantnflem1d  8468  cantnflem1  8469  rexmul  11973  1arithlem4  15468  ramcl  15571  mplcoe1  19286  mplcoe5  19289  subrgascl  19319  scmatscm  20138  marrepfval  20185  ma1repveval  20196  mulmarep1el  20197  mdetralt2  20234  mdetunilem8  20244  maduval  20263  maducoeval2  20265  madurid  20269  minmar1val0  20272  monmatcollpw  20403  pmatcollpwscmatlem1  20413  monmat2matmon  20448  itg2monolem1  23323  iblmulc2  23403  itgmulc2lem1  23404  bddmulibl  23411  dvtaylp  23928  dchrinvcl  24778  rpvmasum2  25001  padicfval  25105  plymulx  29951  itg2addnclem  32631  itg2addnclem3  32633  itg2addnc  32634  itgmulc2nclem1  32646  hdmap1fval  36104  itgioocnicc  38869  etransclem14  39141  etransclem17  39144  etransclem21  39148  etransclem25  39152  etransclem28  39155  etransclem31  39158  hsphoif  39466  hoidmvval  39467  hsphoival  39469  hoidmvlelem5  39489  hoidmvle  39490  ovnhoi  39493  hspmbllem2  39517