Theorem ifnotdc 3641

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 840	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		notnot 632	. . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑)
3	2	iffalsed 3612	. . . 4 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐵)
4		iftrue 3607	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐵)
5	3, 4	eqtr4d 2265	. . 3 ⊢ (𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
6		iftrue 3607	. . . 4 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = 𝐴)
7		iffalse 3610	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐵, 𝐴) = 𝐴)
8	6, 7	eqtr4d 2265	. . 3 ⊢ (¬ 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
9	5, 8	jaoi 721	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))
10	1, 9	sylbi 121	1 ⊢ (DECID 𝜑 → if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 713  DECID wdc 839   = wceq 1395  ifcif 3602

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  lgsneg  15688  lgsdilem  15691