Theorem ifiddc 3591

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 837	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3562	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3		iffalse 3565	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
4	2, 3	jaoi 717	. 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 709  DECID wdc 835   = wceq 1364  ifcif 3557

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-dc 836  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-if 3558

This theorem is referenced by:  xaddpnf1  9912  xaddmnf1  9914  isumz  11532  prod1dc  11729  1arithlem4  12504  xpscf  12930  lgsval2lem  15126  lgsdilem2  15152