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Mirrors > Home > MPE Home > Th. List > difjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-dif 3844. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
difjust | ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2815 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
2 | eleq1w 2815 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)) | |
3 | 2 | notbid 321 | . . . 4 ⊢ (𝑥 = 𝑧 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑧 ∈ 𝐵)) |
4 | 1, 3 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝐵))) |
5 | 4 | cbvabv 2806 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝐵)} |
6 | eleq1w 2815 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
7 | eleq1w 2815 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
8 | 7 | notbid 321 | . . . 4 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵)) |
9 | 6, 8 | anbi12d 634 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵))) |
10 | 9 | cbvabv 2806 | . 2 ⊢ {𝑧 ∣ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)} |
11 | 5, 10 | eqtri 2761 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {cab 2716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 |
This theorem is referenced by: (None) |
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