Theorem ibllem 25733

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 5112	. . . 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 4505	. 2 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0))
5	1	adantrr 718	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → 𝐵 = 𝐶)
6	5	ifeq1da 4513	. 2 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
7	4, 6	eqtrd 2772	1 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ifcif 4481   class class class wbr 5100  0cc0 11038   ≤ cle 11179

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101

This theorem is referenced by:  isibl  25734  isibl2  25735  iblitg  25737  iblcnlem1  25757  iblcnlem  25758  itgcnlem  25759  iblrelem  25760  itgrevallem1  25764  itgeqa  25783