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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 2363. Check out sbcom3vv 2090 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 2504 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑥 / 𝑧]𝜑) | |
2 | sbcom3 2497 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑦 / 𝑥]𝜑) | |
3 | sbcom3 2497 | . 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 205 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: (None) |
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