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Theorem ifbieq12i 4454
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 4432 . . 3 (𝐴 = 𝐶 → if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵))
31, 2ax-mp 5 . 2 if(𝜑, 𝐴, 𝐵) = if(𝜑, 𝐶, 𝐵)
4 ifbieq12i.1 . . 3 (𝜑𝜓)
5 ifbieq12i.3 . . 3 𝐵 = 𝐷
64, 5ifbieq2i 4452 . 2 if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
73, 6eqtri 2824 1 if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  ifcif 4428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-un 3889  df-if 4429
This theorem is referenced by:  cbvditg  24461  sgnneg  31912  binomcxplemdvsum  41056
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