Theorem ifeqda 4507

Description: Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)

Hypotheses

Ref	Expression
ifeqda.1	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
ifeqda.2	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)

Assertion

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

Proof of Theorem ifeqda

Step	Hyp	Ref	Expression
1		iftrue 4476	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐴)
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐴)
3		ifeqda.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
4	2, 3	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶)
5		iffalse 4479	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐵) = 𝐵)
6	5	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐵)
7		ifeqda.2	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)
8	6, 7	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = 𝐶)
9	4, 8	pm2.61dan 812	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ifcif 4470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4471

This theorem is referenced by:  somincom  6076  cantnfp1  9566  ccatsymb  14485  swrdccat3blem  14641  repswccat  14688  ccatco  14737  bitsinvp1  16355  xrsdsreval  21343  fvmptnn04if  22759  chfacfscmulgsum  22770  chfacfpmmulgsum  22774  oprpiece1res2  24872  phtpycc  24912  atantayl2  26870  ifeq3da  32518  fprodex01  32800  psgnfzto1stlem  33061  fzto1st1  33063  cycpm2tr  33080  elrgspnlem4  33204  elrspunsn  33386  fldextrspunlsp  33679  mdetlap1  33831  madjusmdetlem1  33832  madjusmdetlem2  33833  ccatmulgnn0dir  34547  plymulx  34553  itgexpif  34611  repr0  34616  elmrsubrn  35556  matunitlindflem1  37656  sticksstones12  42191  redvmptabs  42393  readvrec  42395  frlmvscadiccat  42539  fsuppind  42623  fsuppssindlem1  42624  reabsifneg  43665  reabsifnpos  43666  reabsifpos  43667  reabsifnneg  43668  reabssgn  43669  sqrtcval  43674  mnringmulrcld  44261  fourierdlem101  46245  hoidmv1lelem2  46630  dfafv2  47163  m1modmmod  47389  linc0scn0  48455  digexp  48639