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Mirrors > Home > MPE Home > Th. List > nfraldwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfraldw 3300 as of 24-Sep-2024. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfraldwOLD.1 | ⊢ Ⅎ𝑦𝜑 |
nfraldwOLD.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfraldwOLD.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfraldwOLD | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3056 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfraldwOLD.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2898 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfraldwOLD.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeld 2908 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfraldwOLD.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfimd 1889 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
8 | 2, 7 | nfald 2315 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
9 | 1, 8 | nfxfrd 1848 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∀wral 3055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 |
This theorem is referenced by: (None) |
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