Theorem ifiddc 3645

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 844	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3614	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3		iffalse 3617	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
4	2, 3	jaoi 724	. 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 716  DECID wdc 842   = wceq 1398  ifcif 3607

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 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-if 3608

This theorem is referenced by:  xaddpnf1  10125  xaddmnf1  10127  isumz  12013  prod1dc  12210  1arithlem4  13002  xpscf  13493  lgsval2lem  15812  lgsdilem2  15838