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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfcf | Structured version Visualization version GIF version |
Description: Version of df-nfc 2740 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.) |
Ref | Expression |
---|---|
bj-nfcf.nf | ⊢ Ⅎ𝑦𝐴 |
Ref | Expression |
---|---|
bj-nfcf | ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2740 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | |
2 | bj-nfcf.nf | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
3 | 2 | nfcri 2745 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 |
4 | 3 | nfnf 2144 | . . . 4 ⊢ Ⅎ𝑦Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | 4 | sb8 2412 | . . 3 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴) |
6 | bj-sbnf 32016 | . . . . 5 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴) | |
7 | clelsb3 2716 | . . . . . 6 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) | |
8 | 7 | nfbii 1770 | . . . . 5 ⊢ (Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴) |
9 | 6, 8 | bitri 263 | . . . 4 ⊢ ([𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴) |
10 | 9 | albii 1737 | . . 3 ⊢ (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
11 | 5, 10 | bitri 263 | . 2 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
12 | 1, 11 | bitri 263 | 1 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 [wsb 1867 ∈ wcel 1977 Ⅎwnfc 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-cleq 2603 df-clel 2606 df-nfc 2740 |
This theorem is referenced by: (None) |
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