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Theorem ibllem 24359
Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
Hypothesis
Ref Expression
ibllem.1 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
ibllem (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))

Proof of Theorem ibllem
StepHypRef Expression
1 ibllem.1 . . . . 5 ((𝜑𝑥𝐴) → 𝐵 = 𝐶)
21breq2d 5061 . . . 4 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ 0 ≤ 𝐶))
32pm5.32da 582 . . 3 (𝜑 → ((𝑥𝐴 ∧ 0 ≤ 𝐵) ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
43ifbid 4470 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
51adantrr 716 . . 3 ((𝜑 ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
65ifeq1da 4478 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
74, 6eqtrd 2859 1 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  ifcif 4448   class class class wbr 5049  0cc0 10524  cle 10663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5050
This theorem is referenced by:  isibl  24360  isibl2  24361  iblitg  24363  iblcnlem1  24382  iblcnlem  24383  itgcnlem  24384  iblrelem  24385  itgrevallem1  24389  itgeqa  24408
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