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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 1473 | . . . 4 ⊢ Ⅎ𝑧𝜑 |
3 | 2 | sbco2yz 1975 | . . 3 ⊢ ([𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑) |
4 | 3 | sbbii 1776 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑤][𝑤 / 𝑥]𝜑) |
5 | nfv 1539 | . . 3 ⊢ Ⅎ𝑤[𝑧 / 𝑥]𝜑 | |
6 | 5 | sbco2yz 1975 | . 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑) |
7 | nfv 1539 | . . 3 ⊢ Ⅎ𝑤𝜑 | |
8 | 7 | sbco2yz 1975 | . 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 1362 [wsb 1773 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: sbco2 1977 sbco2d 1978 sbco3 1986 sb9 1991 elsb1 2167 elsb2 2168 |
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