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Theorem ifbieq2d 3682
 Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
ifbieq2d.1 (φ → (ψχ))
ifbieq2d.2 (φA = B)
Assertion
Ref Expression
ifbieq2d (φ → if(ψ, C, A) = if(χ, C, B))

Proof of Theorem ifbieq2d
StepHypRef Expression
1 ifbieq2d.1 . . 3 (φ → (ψχ))
21ifbid 3680 . 2 (φ → if(ψ, C, A) = if(χ, C, A))
3 ifbieq2d.2 . . 3 (φA = B)
43ifeq2d 3677 . 2 (φ → if(χ, C, A) = if(χ, C, B))
52, 4eqtrd 2385 1 (φ → if(ψ, C, A) = if(χ, C, B))
 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:  tfineq  4488
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