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Mirrors > Home > MPE Home > Th. List > nfsbcw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc 3797 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfsbcw.1 | ⊢ Ⅎ𝑥𝐴 |
nfsbcw.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfsbcw | ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1805 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfsbcw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfsbcw.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfsbcdw 3793 | . 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑) |
7 | 6 | mptru 1544 | 1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1538 Ⅎwnf 1784 Ⅎwnfc 2961 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sbc 3773 |
This theorem is referenced by: opelopabgf 5427 opelopabf 5432 ralrnmptw 6860 elovmporab 7391 elovmporab1w 7392 ovmpt3rabdm 7404 elovmpt3rab1 7405 dfopab2 7750 dfoprab3s 7751 mpoxopoveq 7885 elmptrab 22435 bnj1445 32316 bnj1446 32317 bnj1467 32326 indexa 35023 sdclem1 35033 sbcalf 35407 sbcexf 35408 sbccomieg 39439 rexrabdioph 39440 or2expropbilem2 43317 or2expropbi 43318 ich2exprop 43682 ichnreuop 43683 reuopreuprim 43737 |
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