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Metamath Proof Explorer

Theorem hbn1fw

Description: Weak version of ax-10 from which we can prove any ax-10 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017) (Proof shortened by Wolf Lammen, 28-Feb-2018)

Ref Expression
Hypotheses hbn1fw.1 ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
hbn1fw.2 ⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
hbn1fw.3 ⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
hbn1fw.4 ⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 )
hbn1fw.5 ⊢ ( ¬ ∀ 𝑦 𝜓 → ∀ 𝑥 ¬ ∀ 𝑦 𝜓 )
hbn1fw.6 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion hbn1fw ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 hbn1fw.1 ⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
2 hbn1fw.2 ⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
3 hbn1fw.3 ⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
4 hbn1fw.4 ⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 )
5 hbn1fw.5 ⊢ ( ¬ ∀ 𝑦 𝜓 → ∀ 𝑥 ¬ ∀ 𝑦 𝜓 )
6 hbn1fw.6 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
7 1 2 3 4 6 cbvalw ⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )
8 7 notbii ⊢ ( ¬ ∀ 𝑥 𝜑 ↔ ¬ ∀ 𝑦 𝜓 )
9 8 5 hbxfrbi ⊢ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )