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Theorem ifbi 3497
 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 463 . . . 4 ((𝜑𝜓) → ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓)))
2 notbi 656 . . . . 5 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
32anbi2d 460 . . . 4 ((𝜑𝜓) → ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (𝑥𝐵 ∧ ¬ 𝜓)))
41, 3orbi12d 783 . . 3 ((𝜑𝜓) → (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))))
54abbidv 2258 . 2 ((𝜑𝜓) → {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))})
6 df-if 3480 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
7 df-if 3480 . 2 if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))}
85, 6, 73eqtr4g 2198 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 698   = wceq 1332   ∈ wcel 1481  {cab 2126  ifcif 3479 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-if 3480 This theorem is referenced by:  ifbid  3498  ifbieq2i  3500  fodjuomni  7029  fodjumkv  7042  1tonninf  10245  nninfsellemqall  13387  nninfomni  13391
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