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Metamath Proof Explorer
Description: Weak version of hbn1 . Uses only Tarski's FOL axiom schemes.
(Contributed by NM, 9-Apr-2017)
|
|
Ref |
Expression |
|
Hypothesis |
hbn1w.1 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) |
|
Assertion |
hbn1w |
⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hbn1w.1 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) ) |
| 2 |
|
ax-5 |
⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| 3 |
|
ax-5 |
⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 ) |
| 4 |
|
ax-5 |
⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 ) |
| 5 |
|
ax-5 |
⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 ) |
| 6 |
|
ax-5 |
⊢ ( ¬ ∀ 𝑦 𝜓 → ∀ 𝑥 ¬ ∀ 𝑦 𝜓 ) |
| 7 |
2 3 4 5 6 1
|
hbn1fw |
⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |