Theorem ifeq1da 4516

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 4504	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 4493	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 4493	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2767	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
6	5	adantl 481	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7	2, 6	pm2.61dan 812	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ifcif 4484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-un 3916  df-if 4485

This theorem is referenced by:  ifeq12da  4518  cantnflem1d  9617  cantnflem1  9618  dfac12lem1  10073  xrmaxeq  13115  xrmineq  13116  rexmul  13207  max0add  15252  sumeq2ii  15635  fsumser  15672  ramcl  16976  dmdprdsplitlem  19953  coe1pwmul  22198  scmatscmiddistr  22428  mulmarep1gsum1  22493  maducoeval2  22560  madugsum  22563  madurid  22564  ptcld  23533  ibllem  25698  itgvallem3  25720  iblposlem  25726  iblss2  25740  iblmulc2  25765  cnplimc  25821  limcco  25827  dvexp3  25915  dchrinvcl  27197  lgsval2lem  27251  lgsval4lem  27252  lgsneg  27265  lgsmod  27267  lgsdilem2  27277  rpvmasum2  27456  mrsubrn  35493  ftc1anclem6  37685  ftc1anclem8  37687  fsuppind  42571