![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equsb3r | Structured version Visualization version GIF version |
Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2101. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.) |
Ref | Expression |
---|---|
equsb3r | ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 2023 | . 2 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤)) | |
2 | equequ2 2023 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦)) | |
3 | 1, 2 | sbievw2 2096 | 1 ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 |
This theorem is referenced by: icheq 47387 |
Copyright terms: Public domain | W3C validator |