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Theorem ifbi 3431
 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 456 . . . 4 ((𝜑𝜓) → ((𝑥𝐴𝜑) ↔ (𝑥𝐴𝜓)))
2 id 19 . . . . . 6 ((𝜑𝜓) → (𝜑𝜓))
32notbid 630 . . . . 5 ((𝜑𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))
43anbi2d 453 . . . 4 ((𝜑𝜓) → ((𝑥𝐵 ∧ ¬ 𝜑) ↔ (𝑥𝐵 ∧ ¬ 𝜓)))
51, 4orbi12d 745 . . 3 ((𝜑𝜓) → (((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑)) ↔ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))))
65abbidv 2212 . 2 ((𝜑𝜓) → {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))} = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))})
7 df-if 3414 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
8 df-if 3414 . 2 if(𝜓, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜓) ∨ (𝑥𝐵 ∧ ¬ 𝜓))}
96, 7, 83eqtr4g 2152 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 667   = wceq 1296   ∈ wcel 1445  {cab 2081  ifcif 3413 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-if 3414 This theorem is referenced by:  ifbid  3432  ifbieq2i  3434  fodjuomni  6892  fodjumkv  6903  1tonninf  9995  nninfsellemqall  12616  nninfomni  12620
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