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Theorem ifeq12da 4068
 Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
Hypotheses
Ref Expression
ifeq12da.1 ((𝜑𝜓) → 𝐴 = 𝐶)
ifeq12da.2 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
Assertion
Ref Expression
ifeq12da (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Proof of Theorem ifeq12da
StepHypRef Expression
1 ifeq12da.1 . . . 4 ((𝜑𝜓) → 𝐴 = 𝐶)
21ifeq1da 4066 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3 iftrue 4042 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4 iftrue 4042 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
53, 4eqtr4d 2647 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
62, 5sylan9eq 2664 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7 ifeq12da.2 . . . 4 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
87ifeq2da 4067 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9 iffalse 4045 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10 iffalse 4045 . . . 4 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
119, 10eqtr4d 2647 . . 3 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
128, 11sylan9eq 2664 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
136, 12pm2.61dan 828 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-un 3545  df-if 4037 This theorem is referenced by:  ifbieq12d2  4069
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