Theorem ifeq1dadc 3671

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

Hypotheses

Ref	Expression
ifeq1dadc.1	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
ifeq1dadc.dc	⊢ (𝜑 → DECID 𝜓)

Assertion

Ref	Expression
ifeq1dadc	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1dadc

Step	Hyp	Ref	Expression
1		ifeq1dadc.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
2	1	ifeq1d 3658	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 3648	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 3648	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2274	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
6	5	adantl 277	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7		ifeq1dadc.dc	. . 3 ⊢ (𝜑 → DECID 𝜓)
8		exmiddc 848	. . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
10	2, 6, 9	mpjaodan 810	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402  ifcif 3638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-if 3639

This theorem is referenced by:  sumeq2  12108  isumss  12141  prodeq2  12307  lgsval2lem  16112  lgsval4lem  16113  lgsneg  16126  lgsmod  16128  lgsdilem2  16138