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Theorem ifeq1 3666
 Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
ifeq1 (A = B → if(φ, A, C) = if(φ, B, C))

Proof of Theorem ifeq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 rabeq 2853 . . 3 (A = B → {x A φ} = {x B φ})
21uneq1d 3417 . 2 (A = B → ({x A φ} ∪ {x C ¬ φ}) = ({x B φ} ∪ {x C ¬ φ}))
3 dfif6 3665 . 2 if(φ, A, C) = ({x A φ} ∪ {x C ¬ φ})
4 dfif6 3665 . 2 if(φ, B, C) = ({x B φ} ∪ {x C ¬ φ})
52, 3, 43eqtr4g 2410 1 (A = B → if(φ, A, C) = if(φ, B, C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642  {crab 2618   ∪ cun 3207   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-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663 This theorem is referenced by:  ifeq12  3675  ifeq1d  3676  ifbieq12i  3683  ifexg  3721
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