Theorem spfw 1691

Description: Weak version of sp 1747. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. TO DO: Do we need this theorem? If not, maybe it should be deleted. (Contributed by NM, 19-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
spfw	⊢ (∀xφ → φ)

Distinct variable group:   x,y

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

Proof of Theorem spfw

Step	Hyp	Ref	Expression
1		spfw.2	. . 3 ⊢ (∀xφ → ∀y∀xφ)
2		ax-5 1557	. . 3 ⊢ (∀y(∀xφ → ψ) → (∀y∀xφ → ∀yψ))
3		spfw.3	. . . 4 ⊢ (¬ φ → ∀y ¬ φ)
4		spfw.4	. . . . . 6 ⊢ (x = y → (φ ↔ ψ))
5	4	biimprd 214	. . . . 5 ⊢ (x = y → (ψ → φ))
6	5	equcoms 1681	. . . 4 ⊢ (y = x → (ψ → φ))
7	3, 6	spimw 1668	. . 3 ⊢ (∀yψ → φ)
8	1, 2, 7	syl56 30	. 2 ⊢ (∀y(∀xφ → ψ) → (∀xφ → φ))
9		spfw.1	. . 3 ⊢ (¬ ψ → ∀x ¬ ψ)
10	4	biimpd 198	. . 3 ⊢ (x = y → (φ → ψ))
11	9, 10	spimw 1668	. 2 ⊢ (∀xφ → ψ)
12	8, 11	mpg 1548	1 ⊢ (∀xφ → φ)

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:  spnfwOLD  1692