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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 2371. (Revised by Wolf Lammen, 10-Dec-2019.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfnfc | ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nfc 2886 | . 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴) | |
2 | nfnfc.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | df-nfc 2886 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) | |
4 | 2, 3 | mpbi 233 | . . . . 5 ⊢ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | 4 | spi 2181 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
6 | 5 | nfnf 2325 | . . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴 |
7 | 6 | nfal 2322 | . 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴 |
8 | 1, 7 | nfxfr 1860 | 1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1541 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-nfc 2886 |
This theorem is referenced by: (None) |
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