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Theorem ifbieq12i 3683
 Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
ifbieq12i.1 (φψ)
ifbieq12i.2 A = C
ifbieq12i.3 B = D
Assertion
Ref Expression
ifbieq12i if(φ, A, B) = if(ψ, C, D)

Proof of Theorem ifbieq12i
StepHypRef Expression
1 ifbieq12i.2 . . 3 A = C
2 ifeq1 3666 . . 3 (A = C → if(φ, A, B) = if(φ, C, B))
31, 2ax-mp 8 . 2 if(φ, A, B) = if(φ, C, B)
4 ifbieq12i.1 . . 3 (φψ)
5 ifbieq12i.3 . . 3 B = D
64, 5ifbieq2i 3681 . 2 if(φ, C, B) = if(ψ, C, D)
73, 6eqtri 2373 1 if(φ, A, B) = if(ψ, C, D)
 Colors of variables: wff setvar class Syntax hints:   ↔ 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: (None)
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