Theorem ifiddc 3676

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 848	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3645	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3		iffalse 3648	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
4	2, 3	jaoi 728	. 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 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-11 1559  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-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-if 3639

This theorem is referenced by:  xaddpnf1  10231  xaddmnf1  10233  isumz  12139  prod1dc  12336  1arithlem4  13128  xpscf  13651  lgsval2lem  16112  lgsdilem2  16138