Theorem hba1w 1707

Description: Weak version of hba1 1786. See comments for ax6w 1717. 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
hba1w	⊢ (∀xφ → ∀x∀xφ)

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

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

Proof of Theorem hba1w

Step	Hyp	Ref	Expression
1		hbn1w.1	. . . . . . 7 ⊢ (x = y → (φ ↔ ψ))
2	1	cbvalvw 1702	. . . . . 6 ⊢ (∀xφ ↔ ∀yψ)
3	2	a1i 10	. . . . 5 ⊢ (x = y → (∀xφ ↔ ∀yψ))
4	3	notbid 285	. . . 4 ⊢ (x = y → (¬ ∀xφ ↔ ¬ ∀yψ))
5	4	spw 1694	. . 3 ⊢ (∀x ¬ ∀xφ → ¬ ∀xφ)
6	5	con2i 112	. 2 ⊢ (∀xφ → ¬ ∀x ¬ ∀xφ)
7	4	hbn1w 1706	. 2 ⊢ (¬ ∀x ¬ ∀xφ → ∀x ¬ ∀x ¬ ∀xφ)
8	1	hbn1w 1706	. . . 4 ⊢ (¬ ∀xφ → ∀x ¬ ∀xφ)
9	8	con1i 121	. . 3 ⊢ (¬ ∀x ¬ ∀xφ → ∀xφ)
10	9	alimi 1559	. 2 ⊢ (∀x ¬ ∀x ¬ ∀xφ → ∀x∀xφ)
11	6, 7, 10	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: (None)