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Theorem ibllem 25814
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 5160 . . . 4 ((𝜑𝑥𝐴) → (0 ≤ 𝐵 ↔ 0 ≤ 𝐶))
32pm5.32da 579 . . 3 (𝜑 → ((𝑥𝐴 ∧ 0 ≤ 𝐵) ↔ (𝑥𝐴 ∧ 0 ≤ 𝐶)))
43ifbid 4554 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
51adantrr 717 . . 3 ((𝜑 ∧ (𝑥𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
65ifeq1da 4562 . 2 (𝜑 → if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
74, 6eqtrd 2775 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 1537  wcel 2106  ifcif 4531   class class class wbr 5148  0cc0 11153  cle 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149
This theorem is referenced by:  isibl  25815  isibl2  25816  iblitg  25818  iblcnlem1  25838  iblcnlem  25839  itgcnlem  25840  iblrelem  25841  itgrevallem1  25845  itgeqa  25864
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