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Theorem ifbieq12d 3684
 Description: Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
ifbieq12d.1 (φ → (ψχ))
ifbieq12d.2 (φA = C)
ifbieq12d.3 (φB = D)
Assertion
Ref Expression
ifbieq12d (φ → if(ψ, A, B) = if(χ, C, D))

Proof of Theorem ifbieq12d
StepHypRef Expression
1 ifbieq12d.1 . . 3 (φ → (ψχ))
21ifbid 3680 . 2 (φ → if(ψ, A, B) = if(χ, A, B))
3 ifbieq12d.2 . . 3 (φA = C)
4 ifbieq12d.3 . . 3 (φB = D)
53, 4ifeq12d 3678 . 2 (φ → if(χ, A, B) = if(χ, C, D))
62, 5eqtrd 2385 1 (φ → if(ψ, A, B) = if(χ, C, D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   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:  csbifg  3690  phi11lem1  4595
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