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| 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 2010 | . 2 ⊢ 𝑥 = 𝑥 | |
| 2 | sbequ12r 2251 | . 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 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 | 
| This theorem is referenced by: sbcovOLD 2256 sbco 2511 sbidm 2514 abid 2717 sbceq1a 3798 sbcid 3804 frege58bid 43920 ichid 47443 sbidd 49292 sbidd-misc 49293 | 
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