Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbidm | Structured version Visualization version GIF version |
Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jul-2019.) |
Ref | Expression |
---|---|
sbidm | ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcom3 2411 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑥]𝜑) | |
2 | sbid 2114 | . . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 2 | sbbii 1887 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
4 | 1, 3 | bitr3i 266 | 1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 1880 |
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-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |