Theorem ifnotdc 3555

Description: Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)

Assertion

Ref	Expression
ifnotdc	⊢ (DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))

Proof of Theorem ifnotdc

Step	Hyp	Ref	Expression
1		df-dc 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		notnot 619	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
3	2	iffalsed 3529	. . . 4 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
4		iftrue 3524	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
5	3, 4	eqtr4d 2201	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
6		iftrue 3524	. . . 4 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
7		iffalse 3527	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
8	6, 7	eqtr4d 2201	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
9	5, 8	jaoi 706	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
10	1, 9	sylbi 120	1 ⊢ (DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 698  DECID wdc 824   = wceq 1343  ifcif 3519

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-if 3520

This theorem is referenced by:  lgsneg  13525  lgsdilem  13528