Theorem iftrue 3669

Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem iftrue

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dedlem0a 918	. . 3 ⊢ (φ → (x ∈ A ↔ ((x ∈ B → φ) → (x ∈ A ∧ φ))))
2	1	abbi2dv 2469	. 2 ⊢ (φ → A = {x ∣ ((x ∈ B → φ) → (x ∈ A ∧ φ))})
3		dfif2 3665	. 2 ⊢ if(φ, A, B) = {x ∣ ((x ∈ B → φ) → (x ∈ A ∧ φ))}
4	2, 3	syl6reqr 2404	1 ⊢ (φ → if(φ, A, B) = A)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   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:  ifsb  3672  ifbi  3680  ifeq2da  3689  ifclda  3690  elimif  3692  ifbothda  3693  ifid  3695  ifeqor  3700  ifnot  3701  ifan  3702  ifor  3703  dedth  3704  elimhyp  3711  elimhyp2v  3712  elimhyp3v  3713  elimhyp4v  3714  elimdhyp  3716  keephyp2v  3718  keephyp3v  3719  setswith  4322  dfiota3  4371  eqtfinrelk  4487  tfinnul  4492  dfphi2  4570  phi11lem1  4596  0cnelphi  4598  phialllem1  4617  elimdelov  5574  enprmaplem5  6081