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Mirrors > Home > MPE Home > Th. List > nfnfc | Structured version Visualization version GIF version |
Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2282. (Revised by Wolf Lammen, 10-Dec-2019.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfnfc | ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2782 | . 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴) | |
2 | nfnfc.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcr 2785 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑧 ∈ 𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | 4 | nfnf 2196 | . . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴 |
6 | 5 | nfal 2191 | . 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴 |
7 | 1, 6 | nfxfr 1819 | 1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1521 Ⅎwnf 1748 ∈ wcel 2030 Ⅎwnfc 2780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-11 2074 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ex 1745 df-nf 1750 df-nfc 2782 |
This theorem is referenced by: (None) |
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