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Mirrors > Home > MPE Home > Th. List > sbcom4 | Structured version Visualization version GIF version |
Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 2435 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011.) (Proof shortened by Jim Kingdon, 22-Jan-2018.) |
Ref | Expression |
---|---|
sbcom4 | ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | sbf 2368 | . 2 ⊢ ([𝑤 / 𝑥]𝜑 ↔ 𝜑) |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | sbf 2368 | . . 3 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜑) |
5 | 4 | sbbii 1874 | . 2 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
6 | 3 | sbf 2368 | . . . 4 ⊢ ([𝑤 / 𝑧]𝜑 ↔ 𝜑) |
7 | 6 | sbbii 1874 | . . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
8 | 1 | sbf 2368 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
9 | 7, 8 | bitri 263 | . 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ 𝜑) |
10 | 2, 5, 9 | 3bitr4i 291 | 1 ⊢ ([𝑤 / 𝑥][𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: (None) |
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