Theorem ibllem 25719

Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)

Hypothesis

Ref	Expression
ibllem.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

Assertion

Ref	Expression
ibllem	⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))

Proof of Theorem ibllem

Step	Hyp	Ref	Expression
1		ibllem.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
2	1	breq2d 5108	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ 0 ≤ 𝐶))
3	2	pm5.32da 579	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
4	3	ifbid 4501	. 2 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
5	1	adantrr 717	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
6	5	ifeq1da 4509	. 2 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
7	4, 6	eqtrd 2769	1 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ifcif 4477   class class class wbr 5096  0cc0 11024   ≤ cle 11165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097

This theorem is referenced by:  isibl  25720  isibl2  25721  iblitg  25723  iblcnlem1  25743  iblcnlem  25744  itgcnlem  25745  iblrelem  25746  itgrevallem1  25750  itgeqa  25769