Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbid2vw | Structured version Visualization version GIF version |
Description: Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2551, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.) |
Ref | Expression |
---|---|
sbid2vw | ⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12r 2254 | . 2 ⊢ (𝑥 = 𝑡 → ([𝑥 / 𝑡]𝜑 ↔ 𝜑)) | |
2 | 1 | sbievw 2103 | 1 ⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 |
This theorem is referenced by: sbco4lem 2283 ichid 43660 dfich2bi 43664 |
Copyright terms: Public domain | W3C validator |