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Mirrors > Home > MPE Home > Th. List > nfsbc | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Ref | Expression |
---|---|
nfsbc.1 | ⊢ Ⅎ𝑥𝐴 |
nfsbc.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfsbc | ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1905 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfsbc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfsbc.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfsbcd 3682 | . 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑) |
7 | 6 | mptru 1666 | 1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1659 Ⅎwnf 1884 Ⅎwnfc 2955 [wsbc 3661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-sbc 3662 |
This theorem is referenced by: cbvralcsf 3788 opelopabgf 5220 opelopabf 5225 ralrnmpt 6616 elovmpt2rab 7139 elovmpt2rab1 7140 ovmpt3rabdm 7151 elovmpt3rab1 7152 dfopab2 7483 dfoprab3s 7484 mpt2xopoveq 7609 elmptrab 22000 bnj1445 31657 bnj1446 31658 bnj1467 31667 indexa 34070 sdclem1 34080 sbcalf 34457 sbcexf 34458 sbccomieg 38200 rexrabdioph 38201 |
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