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Theorem ifbieq12d 3348
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 3343 . 2 (φ → if(ψ, A, B) = if(χ, A, B))
3 ifbieq12d.2 . . 3 (φA = 𝐶)
4 ifbieq12d.3 . . 3 (φB = 𝐷)
53, 4ifeq12d 3341 . 2 (φ → if(χ, A, B) = if(χ, 𝐶, 𝐷))
62, 5eqtrd 2069 1 (φ → if(ψ, A, B) = if(χ, 𝐶, 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  ifcif 3325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-un 2916  df-if 3326
This theorem is referenced by:  expival  8871
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