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Theorem ifbieq12i 4252
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 4230 . . 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 4250 . 2 if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
73, 6eqtri 2793 1 if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  ifcif 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-un 3729  df-if 4227
This theorem is referenced by:  cbvditg  23839  sgnneg  30943  binomcxplemdvsum  39081
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