Theorem iffalse 3669

Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)

Assertion

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

Proof of Theorem iffalse

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedlemb 921	. . 3 ⊢ (¬ φ → (x ∈ B ↔ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))))
2	1	abbi2dv 2468	. 2 ⊢ (¬ φ → B = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))})
3		df-if 3663	. 2 ⊢ if(φ, A, B) = {x ∣ ((x ∈ A ∧ φ) ∨ (x ∈ B ∧ ¬ φ))}
4	2, 3	syl6reqr 2404	1 ⊢ (¬ φ → if(φ, A, B) = B)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ifcif 3662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 3663

This theorem is referenced by:  ifnefalse  3670  ifsb  3671  ifbi  3679  ifeq1da  3687  ifclda  3689  elimif  3691  ifbothda  3692  ifid  3694  ifeqor  3699  ifnot  3700  ifan  3701  ifor  3702  elimhyp  3710  elimhyp2v  3711  elimhyp3v  3712  elimhyp4v  3713  elimdhyp  3715  keephyp2v  3717  keephyp3v  3718  setswith  4321  dfiota3  4370  eqtfinrelk  4486  tfinprop  4489  dfphi2  4569  phi11lem1  4595  0cnelphi  4597  phidisjnn  4615  elimdelov  5573  enprmaplem5  6080