![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > vjust | GIF version |
Description: Soundness justification theorem for df-v 2739. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Ref | Expression |
---|---|
vjust | ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1701 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | sbt 1784 | . . . 4 ⊢ [𝑧 / 𝑥]𝑥 = 𝑥 |
3 | equid 1701 | . . . . 5 ⊢ 𝑦 = 𝑦 | |
4 | 3 | sbt 1784 | . . . 4 ⊢ [𝑧 / 𝑦]𝑦 = 𝑦 |
5 | 2, 4 | 2th 174 | . . 3 ⊢ ([𝑧 / 𝑥]𝑥 = 𝑥 ↔ [𝑧 / 𝑦]𝑦 = 𝑦) |
6 | df-clab 2164 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ [𝑧 / 𝑥]𝑥 = 𝑥) | |
7 | df-clab 2164 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦} ↔ [𝑧 / 𝑦]𝑦 = 𝑦) | |
8 | 5, 6, 7 | 3bitr4i 212 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝑥 = 𝑥} ↔ 𝑧 ∈ {𝑦 ∣ 𝑦 = 𝑦}) |
9 | 8 | eqriv 2174 | 1 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦} |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 [wsb 1762 ∈ wcel 2148 {cab 2163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |