Theorem hbn1fw 1705

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, 19-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
hbn1fw	⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)

Distinct variable group:   x,y

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

Proof of Theorem hbn1fw

Step	Hyp	Ref	Expression
1		hbn1fw.1	. . . . 5 ⊢ (∀xφ → ∀y∀xφ)
2		hbn1fw.2	. . . . 5 ⊢ (¬ ψ → ∀x ¬ ψ)
3		hbn1fw.3	. . . . 5 ⊢ (∀yψ → ∀x∀yψ)
4		hbn1fw.4	. . . . 5 ⊢ (¬ φ → ∀y ¬ φ)
5		hbn1fw.6	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvalw 1701	. . . 4 ⊢ (∀xφ ↔ ∀yψ)
7	6	biimpri 197	. . 3 ⊢ (∀yψ → ∀xφ)
8	7	con3i 127	. 2 ⊢ (¬ ∀xφ → ¬ ∀yψ)
9		hbn1fw.5	. 2 ⊢ (¬ ∀yψ → ∀x ¬ ∀yψ)
10	6	biimpi 186	. . . 4 ⊢ (∀xφ → ∀yψ)
11	10	con3i 127	. . 3 ⊢ (¬ ∀yψ → ¬ ∀xφ)
12	11	alimi 1559	. 2 ⊢ (∀x ¬ ∀yψ → ∀x ¬ ∀xφ)
13	8, 9, 12	3syl 18	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:  hbn1w  1706