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Mirrors > Home > MPE Home > Th. List > hbn1w | Structured version Visualization version GIF version |
Description: Weak version of hbn1 2007. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) |
Ref | Expression |
---|---|
hbn1w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
hbn1w | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1827 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
2 | ax-5 1827 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
3 | ax-5 1827 | . 2 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) | |
4 | ax-5 1827 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
5 | ax-5 1827 | . 2 ⊢ (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓) | |
6 | hbn1w.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | 1, 2, 3, 4, 5, 6 | hbn1fw 1959 | 1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∀wal 1473 |
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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: hba1w 1961 hba1wOLD 1962 hbe1w 1963 ax10w 1993 wl-naevhba1v 32483 |
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