Theorem hbn1w 1706

Description: Weak version of hbn1 1730. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
hbn1w.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
hbn1w	⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Distinct variable groups:   φ,y   ψ,x   x,y

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem hbn1w

Step	Hyp	Ref	Expression
1		ax-17 1616	. 2 ⊢ (∀xφ → ∀y∀xφ)
2		ax-17 1616	. 2 ⊢ (¬ ψ → ∀x ¬ ψ)
3		ax-17 1616	. 2 ⊢ (∀yψ → ∀x∀yψ)
4		ax-17 1616	. 2 ⊢ (¬ φ → ∀y ¬ φ)
5		ax-17 1616	. 2 ⊢ (¬ ∀yψ → ∀x ¬ ∀yψ)
6		hbn1w.1	. 2 ⊢ (x = y → (φ ↔ ψ))
7	1, 2, 3, 4, 5, 6	hbn1fw 1705	1 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  hba1w  1707  hbe1w  1708  ax6w  1717