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Mirrors > Home > MPE Home > Th. List > equsb1v | Structured version Visualization version GIF version |
Description: Substitution applied to an atomic wff. Version of equsb1 2494 with a disjoint variable condition, which neither requires ax-12 2177 nor ax-13 2371. (Contributed by NM, 10-May-1993.) (Revised by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise df-sb 2073. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.) |
Ref | Expression |
---|---|
equsb1v | ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2022 | . 2 ⊢ 𝑦 = 𝑦 | |
2 | equsb3 2107 | . 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 ↔ 𝑦 = 𝑦) | |
3 | 1, 2 | mpbir 234 | 1 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 |
Colors of variables: wff setvar class |
Syntax hints: [wsb 2072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2073 |
This theorem is referenced by: pm13.183 3565 exss 5332 |
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