Theorem ifeq1da 4513

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 4501	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 4490	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 4490	. . . 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 481	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7	2, 6	pm2.61dan 813	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ifcif 4481

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-un 3908  df-if 4482

This theorem is referenced by:  ifeq12da  4515  cantnflem1d  9609  cantnflem1  9610  dfac12lem1  10066  xrmaxeq  13106  xrmineq  13107  rexmul  13198  max0add  15245  sumeq2ii  15628  fsumser  15665  ramcl  16969  dmdprdsplitlem  19980  coe1pwmul  22233  scmatscmiddistr  22464  mulmarep1gsum1  22529  maducoeval2  22596  madugsum  22599  madurid  22600  ptcld  23569  ibllem  25733  itgvallem3  25755  iblposlem  25761  iblss2  25775  iblmulc2  25800  cnplimc  25856  limcco  25862  dvexp3  25950  dchrinvcl  27232  lgsval2lem  27286  lgsval4lem  27287  lgsneg  27300  lgsmod  27302  lgsdilem2  27312  rpvmasum2  27491  esplyind  33752  mrsubrn  35729  ftc1anclem6  37949  ftc1anclem8  37951  fsuppind  42948