![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opeq1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of opeq1 4754 as of 25-May-2024. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opeq1OLD | ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2838 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
2 | 1 | anbi1d 633 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ V ∧ 𝐶 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐶 ∈ V))) |
3 | sneq 4525 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵}) | |
4 | preq1 4619 | . . . 4 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶}) | |
5 | 3, 4 | preq12d 4627 | . . 3 ⊢ (𝐴 = 𝐵 → {{𝐴}, {𝐴, 𝐶}} = {{𝐵}, {𝐵, 𝐶}}) |
6 | 2, 5 | ifbieq1d 4437 | . 2 ⊢ (𝐴 = 𝐵 → if((𝐴 ∈ V ∧ 𝐶 ∈ V), {{𝐴}, {𝐴, 𝐶}}, ∅) = if((𝐵 ∈ V ∧ 𝐶 ∈ V), {{𝐵}, {𝐵, 𝐶}}, ∅)) |
7 | dfopif 4750 | . 2 ⊢ 〈𝐴, 𝐶〉 = if((𝐴 ∈ V ∧ 𝐶 ∈ V), {{𝐴}, {𝐴, 𝐶}}, ∅) | |
8 | dfopif 4750 | . 2 ⊢ 〈𝐵, 𝐶〉 = if((𝐵 ∈ V ∧ 𝐶 ∈ V), {{𝐵}, {𝐵, 𝐶}}, ∅) | |
9 | 6, 7, 8 | 3eqtr4g 2819 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐴, 𝐶〉 = 〈𝐵, 𝐶〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∅c0 4221 ifcif 4413 {csn 4515 {cpr 4517 〈cop 4521 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2176 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-dif 3857 df-un 3859 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |