Theorem ifswap 2379

Description: Negating the first argument swaps the last two arguments of a conditional operator.

Assertion

Ref	Expression
ifswap	⊢ if(¬ φ, A, B) = if(φ, B, A)

Proof of Theorem ifswap

Step	Hyp	Ref	Expression
1		negb 86	. . . 4 ⊢ (φ → ¬ ¬ φ)
2		iffalse 2364	. . . 4 ⊢ (¬ ¬ φ → if(¬ φ, A, B) = B)
3	1, 2	syl 10	. . 3 ⊢ (φ → if(¬ φ, A, B) = B)
4		iftrue 2363	. . 3 ⊢ (φ → if(φ, B, A) = B)
5	3, 4	eqtr4d 1508	. 2 ⊢ (φ → if(¬ φ, A, B) = if(φ, B, A))
6		iftrue 2363	. . 3 ⊢ (¬ φ → if(¬ φ, A, B) = A)
7		iffalse 2364	. . 3 ⊢ (¬ φ → if(φ, B, A) = A)
8	6, 7	eqtr4d 1508	. 2 ⊢ (¬ φ → if(¬ φ, A, B) = if(φ, B, A))
9	5, 8	pm2.61i 126	1 ⊢ if(¬ φ, A, B) = if(φ, B, A)

Syntax hints:  ¬ wn 2   = wceq 955   ifcif 2358

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-if 2359

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