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Mirrors > Home > ILE Home > Th. List > sbco2 | GIF version |
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
sbco2.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2 | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfri 1452 | . 2 ⊢ (𝜑 → ∀𝑧𝜑) |
3 | 2 | sbco2h 1879 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 Ⅎwnf 1389 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: nfsbt 1891 sb7af 1910 sbco4lem 1923 sbco4 1924 eqsb3 2182 clelsb3 2183 clelsb4 2184 sb8ab 2200 sbralie 2590 sbcco 2836 |
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