|   | 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 2514, 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 2252 | . 2 ⊢ (𝑥 = 𝑡 → ([𝑥 / 𝑡]𝜑 ↔ 𝜑)) | |
| 2 | 1 | sbievw 2093 | 1 ⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 [wsb 2064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 | 
| This theorem is referenced by: ichid 47438 | 
| Copyright terms: Public domain | W3C validator |