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| Mirrors > Home > MPE Home > Th. List > sbcom | Structured version Visualization version GIF version | ||
| Description: A commutativity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. Check out sbcom3vv 2097 for a version requiring fewer axioms. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-2018.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| sbcom | ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbco3 2518 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑥 / 𝑧]𝜑) | |
| 2 | sbcom3 2511 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑) | |
| 3 | sbcom3 2511 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]𝜑) | |
| 4 | 1, 2, 3 | 3bitr3i 301 | 1 ⊢ ([𝑦 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑦 / 𝑧]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-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |