Theorem ifeq1da 4490

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 4478	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 4468	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 4468	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2781	. . 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 810	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539  ifcif 4459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-un 3892  df-if 4460

This theorem is referenced by:  ifeq12da  4492  cantnflem1d  9446  cantnflem1  9447  dfac12lem1  9899  xrmaxeq  12913  xrmineq  12914  rexmul  13005  max0add  15022  sumeq2ii  15405  fsumser  15442  ramcl  16730  dmdprdsplitlem  19640  coe1pwmul  21450  scmatscmiddistr  21657  mulmarep1gsum1  21722  maducoeval2  21789  madugsum  21792  madurid  21793  ptcld  22764  ibllem  24929  itgvallem3  24950  iblposlem  24956  iblss2  24970  iblmulc2  24995  cnplimc  25051  limcco  25057  dvexp3  25142  dchrinvcl  26401  lgsval2lem  26455  lgsval4lem  26456  lgsneg  26469  lgsmod  26471  lgsdilem2  26481  rpvmasum2  26660  mrsubrn  33475  ftc1anclem6  35855  ftc1anclem8  35857  fsuppind  40279