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Theorem ifeq12da 4308
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 4306 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3 iftrue 4282 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4 iftrue 4282 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
53, 4eqtr4d 2835 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
62, 5sylan9eq 2852 . 2 ((𝜑𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7 ifeq12da.2 . . . 4 ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
87ifeq2da 4307 . . 3 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9 iffalse 4285 . . . 4 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10 iffalse 4285 . . . 4 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
119, 10eqtr4d 2835 . . 3 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
128, 11sylan9eq 2852 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
136, 12pm2.61dan 848 1 (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  ifcif 4276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-rab 3097  df-v 3386  df-un 3773  df-if 4277
This theorem is referenced by:  ifbieq12d2  4309
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