Theorem ifeq1da 4559

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
ifeq1da.1	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)

Assertion

Ref	Expression
ifeq1da	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1da

Step	Hyp	Ref	Expression
1		ifeq1da.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
2	1	ifeq1d 4547	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 4537	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 4537	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2775	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
6	5	adantl 482	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7	2, 6	pm2.61dan 811	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-un 3953  df-if 4529

This theorem is referenced by:  ifeq12da  4561  cantnflem1d  9685  cantnflem1  9686  dfac12lem1  10140  xrmaxeq  13162  xrmineq  13163  rexmul  13254  max0add  15261  sumeq2ii  15643  fsumser  15680  ramcl  16966  dmdprdsplitlem  19948  coe1pwmul  22021  scmatscmiddistr  22230  mulmarep1gsum1  22295  maducoeval2  22362  madugsum  22365  madurid  22366  ptcld  23337  ibllem  25506  itgvallem3  25527  iblposlem  25533  iblss2  25547  iblmulc2  25572  cnplimc  25628  limcco  25634  dvexp3  25719  dchrinvcl  26980  lgsval2lem  27034  lgsval4lem  27035  lgsneg  27048  lgsmod  27050  lgsdilem2  27060  rpvmasum2  27239  mrsubrn  34790  ftc1anclem6  36869  ftc1anclem8  36871  fsuppind  41464