Theorem ifiddc 3607

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 838	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3577	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3		iffalse 3580	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
4	2, 3	jaoi 718	. 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 710  DECID wdc 836   = wceq 1373  ifcif 3572

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-11 1530  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-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-if 3573

This theorem is referenced by:  xaddpnf1  9975  xaddmnf1  9977  isumz  11744  prod1dc  11941  1arithlem4  12733  xpscf  13223  lgsval2lem  15531  lgsdilem2  15557