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Theorem ifbi 3462
Description: Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
Assertion
Ref Expression
ifbi ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Proof of Theorem ifbi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 anbi2 462 . . . 4 ((𝜑𝜓) → ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓)))
2 notbi 640 . . . . 5 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
32anbi2d 459 . . . 4 ((𝜑𝜓) → ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (𝑥𝐵 ∧ ¬ 𝜓)))
41, 3orbi12d 767 . . 3 ((𝜑𝜓) → (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))))
54abbidv 2235 . 2 ((𝜑𝜓) → {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))})
6 df-if 3445 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
7 df-if 3445 . 2 if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))}
85, 6, 73eqtr4g 2175 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682   = wceq 1316  wcel 1465  {cab 2103  ifcif 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-if 3445
This theorem is referenced by:  ifbid  3463  ifbieq2i  3465  fodjuomni  6989  fodjumkv  7002  1tonninf  10181  nninfsellemqall  13138  nninfomni  13142
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