|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sbid | Structured version Visualization version GIF version | ||
| Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| sbid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | equid 2011 | . 2 ⊢ 𝑥 = 𝑥 | |
| 2 | sbequ12r 2252 | . 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑥 / 𝑥]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: sbcovOLD 2257 sbco 2512 sbidm 2515 abid 2718 sbceq1a 3799 sbcid 3805 frege58bid 43915 ichid 47438 sbidd 49237 sbidd-misc 49238 | 
| Copyright terms: Public domain | W3C validator |