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Mirrors > Home > MPE Home > Th. List > vjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-v 3351. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Ref | Expression |
---|---|
vjust | ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2096 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | sbt 2565 | . . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥 |
3 | equid 2096 | . . . . 5 ⊢ 𝑦 = 𝑦 | |
4 | 3 | sbt 2565 | . . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦 |
5 | 2, 4 | 2th 254 | . . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦) |
6 | df-clab 2757 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥) | |
7 | df-clab 2757 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦) | |
8 | 5, 6, 7 | 3bitr4i 292 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦}) |
9 | 8 | eqriv 2767 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1630 [wsb 2048 ∈ wcel 2144 {cab 2756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1852 df-sb 2049 df-clab 2757 df-cleq 2763 |
This theorem is referenced by: (None) |
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