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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 1458 | . 2 ⊢ (𝜑 → ∀𝑧𝜑) |
3 | 2 | sbco2h 1887 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 Ⅎwnf 1395 [wsb 1693 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-nf 1396 df-sb 1694 |
This theorem is referenced by: nfsbt 1899 sb7af 1918 sbco4lem 1931 sbco4 1932 eqsb3 2192 clelsb3 2193 clelsb4 2194 sb8ab 2210 clelsb3f 2233 sbralie 2604 sbcco 2862 |
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