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Mirrors > Home > MPE Home > Th. List > vjust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-v 3439. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Ref | Expression |
---|---|
vjust | ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1996 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | sbt 2044 | . . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥 |
3 | equid 1996 | . . . . 5 ⊢ 𝑦 = 𝑦 | |
4 | 3 | sbt 2044 | . . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦 |
5 | 2, 4 | 2th 265 | . . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦) |
6 | df-clab 2776 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥) | |
7 | df-clab 2776 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦) | |
8 | 5, 6, 7 | 3bitr4i 304 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦}) |
9 | 8 | eqriv 2792 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 [wsb 2042 ∈ wcel 2081 {cab 2775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-9 2091 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-clab 2776 df-cleq 2788 |
This theorem is referenced by: (None) |
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