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Mirrors > Home > ILE Home > Th. List > sbco2yz | GIF version |
Description: This is a version of sbco2 1958 where 𝑧 is distinct from 𝑦. It is a lemma on the way to proving sbco2 1958 which has no distinct variable constraints. (Contributed by Jim Kingdon, 19-Mar-2018.) |
Ref | Expression |
---|---|
sbco2yz.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2yz | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2yz.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsb 1939 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
3 | 2 | nfri 1512 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
4 | sbequ 1833 | . 2 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
5 | 3, 4 | sbieh 1783 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 Ⅎwnf 1453 [wsb 1755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 |
This theorem is referenced by: sbco2h 1957 |
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