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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vjust2 | Structured version Visualization version GIF version |
Description: Justification theorem for bj-df-v 33340. See also vjust 3341 and bj-vjust 33114. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vjust2 | ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2747 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ [𝑧 / 𝑥]⊤) | |
2 | bj-sbfvv 33093 | . . . 4 ⊢ ([𝑧 / 𝑦]⊤ ↔ ⊤) | |
3 | df-clab 2747 | . . . 4 ⊢ (𝑧 ∈ {𝑦 ∣ ⊤} ↔ [𝑧 / 𝑦]⊤) | |
4 | bj-sbfvv 33093 | . . . 4 ⊢ ([𝑧 / 𝑥]⊤ ↔ ⊤) | |
5 | 2, 3, 4 | 3bitr4ri 293 | . . 3 ⊢ ([𝑧 / 𝑥]⊤ ↔ 𝑧 ∈ {𝑦 ∣ ⊤}) |
6 | 1, 5 | bitri 264 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ ⊤} ↔ 𝑧 ∈ {𝑦 ∣ ⊤}) |
7 | 6 | eqriv 2757 | 1 ⊢ {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ⊤wtru 1633 [wsb 2046 ∈ wcel 2139 {cab 2746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-12 2196 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1854 df-sb 2047 df-clab 2747 df-cleq 2753 |
This theorem is referenced by: (None) |
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