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Mirrors > Home > ILE Home > Th. List > nfsbcw | GIF version |
Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3006 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfsbcw.1 | ⊢ Ⅎ𝑥𝐴 |
nfsbcw.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfsbcw | ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1477 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfsbcw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfsbcw.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfsbcdw 3114 | . 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑) |
7 | 6 | mptru 1373 | 1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1365 Ⅎwnf 1471 Ⅎwnfc 2323 [wsbc 2985 |
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-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-sbc 2986 |
This theorem is referenced by: elovmporab 6110 elovmporab1w 6111 |
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