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Theorem ibllem 25145
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 5122 . . . 4 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ 0 ≤ 𝐶))
32pm5.32da 580 . . 3 (𝜑 → ((𝑥𝐴 ∧ 0 ≤ 𝐵) ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
43ifbid 4514 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
51adantrr 716 . . 3 ((𝜑 ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
65ifeq1da 4522 . 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 397   = wceq 1542  wcel 2107  ifcif 4491   class class class wbr 5110  0cc0 11058  cle 11197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111
This theorem is referenced by:  isibl  25146  isibl2  25147  iblitg  25149  iblcnlem1  25168  iblcnlem  25169  itgcnlem  25170  iblrelem  25171  itgrevallem1  25175  itgeqa  25194
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