Theorem ifeq1dadc 3603

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 3590	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 3581	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 3581	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2242	. . 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 838	. . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
10	2, 6, 9	mpjaodan 800	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  DECID wdc 836   = wceq 1373  ifcif 3573

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 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-un 3172  df-if 3574

This theorem is referenced by:  sumeq2  11720  isumss  11752  prodeq2  11918  lgsval2lem  15537  lgsval4lem  15538  lgsneg  15551  lgsmod  15553  lgsdilem2  15563