Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hbntOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of hbnt 2144 as of 13-Oct-2021. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbntOLD | ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ex 1705 | . . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) | |
2 | 19.9ht 2143 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
3 | 1, 2 | syl5bir 233 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥 ¬ 𝜑 → 𝜑)) |
4 | 3 | con1d 139 | 1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-ex 1705 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |