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Theorem ifbi 3569
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 467 . . . 4 ((𝜑𝜓) → ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓)))
2 notbi 667 . . . . 5 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
32anbi2d 464 . . . 4 ((𝜑𝜓) → ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (𝑥𝐵 ∧ ¬ 𝜓)))
41, 3orbi12d 794 . . 3 ((𝜑𝜓) → (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))))
54abbidv 2307 . 2 ((𝜑𝜓) → {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))})
6 df-if 3550 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
7 df-if 3550 . 2 if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))}
85, 6, 73eqtr4g 2247 1 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wcel 2160  {cab 2175  ifcif 3549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-if 3550
This theorem is referenced by:  ifbid  3570  ifbieq2i  3572  fodjuomni  7177  fodjumkv  7188  nninfwlpoimlemg  7203  1tonninf  10471  lgsdi  14899  nninfsellemqall  15226  nninfomni  15230
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