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Theorem ibllem 25799
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 5155 . . . 4 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ 0 ≤ 𝐶))
32pm5.32da 579 . . 3 (𝜑 → ((𝑥𝐴 ∧ 0 ≤ 𝐵) ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
43ifbid 4549 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
51adantrr 717 . . 3 ((𝜑 ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
65ifeq1da 4557 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
74, 6eqtrd 2777 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 395   = wceq 1540  wcel 2108  ifcif 4525   class class class wbr 5143  0cc0 11155  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144
This theorem is referenced by:  isibl  25800  isibl2  25801  iblitg  25803  iblcnlem1  25823  iblcnlem  25824  itgcnlem  25825  iblrelem  25826  itgrevallem1  25830  itgeqa  25849
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