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| Mirrors > Home > MPE Home > Th. List > ibllem | Structured version Visualization version GIF version | ||
| Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.) |
| Ref | Expression |
|---|---|
| ibllem.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| ibllem | ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ibllem.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
| 2 | 1 | breq2d 5155 | . . . 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 4549 | . 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 4557 | . 2 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) |
| 7 | 4, 6 | eqtrd 2777 | 1 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 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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