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Mirrors > Home > MPE Home > Th. List > nfaldOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of nfald 2151 as of 16-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfald.1 | ⊢ Ⅎ𝑦𝜑 |
nfald.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfaldOLD | ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfald.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfald.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 1, 2 | alrimi 2069 | . 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓) |
4 | nfnf1 2018 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓 | |
5 | 4 | nfal 2139 | . . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓 |
6 | hba1 2137 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓) | |
7 | sp 2041 | . . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
8 | 7 | nf5rd 2054 | . . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓)) |
9 | 6, 8 | hbald 2028 | . . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)) |
10 | 5, 9 | nf5d 2104 | . 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓) |
11 | 3, 10 | syl 17 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 Ⅎwnf 1699 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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