Theorem ifid 3695

Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)

Assertion

Ref	Expression
ifid	⊢ if(φ, A, A) = A

Proof of Theorem ifid

Step	Hyp	Ref	Expression
1		iftrue 3669	. 2 ⊢ (φ → if(φ, A, A) = A)
2		iffalse 3670	. 2 ⊢ (¬ φ → if(φ, A, A) = A)
3	1, 2	pm2.61i 156	1 ⊢ if(φ, A, A) = A

Syntax hints:   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by: (None)