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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-vjust | Structured version Visualization version GIF version |
Description: Remove dependency on ax-13 2246 from vjust 3201 (note the absence of DV conditions). Soundness justification theorem for df-v 3202. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vjust | ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1939 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | bj-sbtv 32766 | . . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥 |
3 | equid 1939 | . . . . 5 ⊢ 𝑦 = 𝑦 | |
4 | 3 | bj-sbtv 32766 | . . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦 |
5 | 2, 4 | 2th 254 | . . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦) |
6 | df-clab 2609 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥) | |
7 | df-clab 2609 | . . 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 2619 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 [wsb 1880 ∈ wcel 1990 {cab 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 df-clab 2609 df-cleq 2615 |
This theorem is referenced by: (None) |
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