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Theorem ifbi 3375
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 448 . . . 4 ((𝜑𝜓) → ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓)))
2 id 19 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
32notbid 602 . . . . 5 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43anbi2d 445 . . . 4 ((𝜑𝜓) → ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (𝑥𝐵 ∧ ¬ 𝜓)))
51, 4orbi12d 717 . . 3 ((𝜑𝜓) → (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))))
65abbidv 2171 . 2 ((𝜑𝜓) → {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))})
7 df-if 3359 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
8 df-if 3359 . 2 if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))}
96, 7, 83eqtr4g 2113 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639   = wceq 1259  wcel 1409  {cab 2042  ifcif 3358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-if 3359
This theorem is referenced by:  ifbid  3376  ifbieq2i  3378
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