Theorem ifiddc 3538

Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Assertion

Ref	Expression
ifiddc	⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)

Proof of Theorem ifiddc

Step	Hyp	Ref	Expression
1		exmiddc 822	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3510	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3		iffalse 3513	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
4	2, 3	jaoi 706	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → if(𝜑, 𝐴, 𝐴) = 𝐴)
5	1, 4	syl 14	1 ⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 820   = wceq 1335  ifcif 3505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-if 3506

This theorem is referenced by:  xaddpnf1  9750  xaddmnf1  9752  isumz  11286  prod1dc  11483