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Mirrors > Home > ILE Home > Th. List > sbco2h | GIF version |
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Ref | Expression |
---|---|
sbco2h.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
sbco2h | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2h.1 | . . . . 5 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | nfi 1462 | . . . 4 ⊢ Ⅎ𝑧𝜑 |
3 | 2 | sbco2yz 1963 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
4 | 3 | sbbii 1765 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
5 | nfv 1528 | . . 3 ⊢ Ⅎ𝑤[𝑧 / 𝑥]𝜑 | |
6 | 5 | sbco2yz 1963 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
7 | nfv 1528 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
8 | 7 | sbco2yz 1963 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
9 | 4, 6, 8 | 3bitr3i 210 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 [wsb 1762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 |
This theorem is referenced by: sbco2 1965 sbco2d 1966 sbco3 1974 sb9 1979 elsb1 2155 elsb2 2156 |
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