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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vjust | Structured version Visualization version GIF version |
Description: Justification theorem for bj-df-v 34471. See also vjust 3442. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vjust | ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2777 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤) | |
2 | sbv 2095 | . . . 4 ⊢ ([𝑧 / 𝑦]⊤ ↔ ⊤) | |
3 | df-clab 2777 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤) | |
4 | sbv 2095 | . . . 4 ⊢ ([𝑧 / 𝑥]⊤ ↔ ⊤) | |
5 | 2, 3, 4 | 3bitr4ri 307 | . . 3 ⊢ ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤}) |
6 | 1, 5 | bitri 278 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤}) |
7 | 6 | eqriv 2795 | 1 ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ⊤wtru 1539 [wsb 2069 ∈ wcel 2111 {cab 2776 |
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 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 |
This theorem is referenced by: (None) |
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