Theorem ax6w 1717

Description: Weak version of ax-6 1729 from which we can prove any ax-6 1729 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
ax6w.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
ax6w	⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

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

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

Proof of Theorem ax6w

Step	Hyp	Ref	Expression
1		ax6w.1	. 2 ⊢ (x = y → (φ ↔ ψ))
2	1	hbn1w 1706	1 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540

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

This theorem is used by: (None)