Description: Weak version of hba1 . See comments for ax10w . Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017) (Proof shortened by Wolf Lammen, 10-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | hbn1w.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| Assertion | hba1w | ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hbn1w.1 | ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) | |
| 2 | 1 | cbvalvw | ⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 ) |
| 3 | 2 | notbii | ⊢ ( ¬ ∀ 𝑥 𝜑 ↔ ¬ ∀ 𝑦 𝜓 ) |
| 4 | 3 | a1i | ⊢ ( 𝑥 = 𝑦 → ( ¬ ∀ 𝑥 𝜑 ↔ ¬ ∀ 𝑦 𝜓 ) ) |
| 5 | 4 | spw | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 𝜑 ) |
| 6 | 5 | con2i | ⊢ ( ∀ 𝑥 𝜑 → ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 7 | 4 | hbn1w | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 8 | 1 | hbn1w | ⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 9 | 8 | con1i | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) |
| 10 | 9 | alimi | ⊢ ( ∀ 𝑥 ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) |
| 11 | 6 7 10 | 3syl | ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑥 𝜑 ) |