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Mirrors > Home > ILE Home > Th. List > nfsb | GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.) |
Ref | Expression |
---|---|
nfsb.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
nfsb | ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsb.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfsbxy 1954 | . . 3 ⊢ Ⅎ𝑧[𝑤 / 𝑥]𝜑 |
3 | 2 | nfsbxy 1954 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 |
4 | ax-17 1537 | . . . 4 ⊢ (𝜑 → ∀𝑤𝜑) | |
5 | 4 | sbco2vh 1957 | . . 3 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
6 | 5 | nfbii 1484 | . 2 ⊢ (Ⅎ𝑧[𝑦 / 𝑤][𝑤 / 𝑥]𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
7 | 3, 6 | mpbi 145 | 1 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1471 [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: hbsb 1961 sbco2yz 1975 sbcomxyyz 1984 hbsbd 1994 nfsb4or 2033 sb8eu 2051 nfeu 2057 cbvab 2313 cbvralf 2710 cbvrexf 2711 cbvreu 2716 cbvralsv 2734 cbvrexsv 2735 cbvrab 2750 cbvreucsf 3136 cbvrabcsf 3137 cbvopab1 4091 cbvmptf 4112 cbvmpt 4113 ralxpf 4791 rexxpf 4792 cbviota 5201 sb8iota 5203 cbvriota 5862 dfoprab4f 6218 |
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