Theorem spnfwOLD 1692

Description: Weak version of sp 1747. Uses only Tarski's FOL axiom schemes. Obsolete version of spnfw 1670 as of 13-Aug-2017. (Contributed by NM, 1-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
spnfw.3	⊢ (¬ φ → ∀x ¬ φ)

Assertion

Ref	Expression
spnfwOLD	⊢ (∀xφ → φ)

Proof of Theorem spnfwOLD

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spnfw.3	. 2 ⊢ (¬ φ → ∀x ¬ φ)
2		ax-17 1616	. 2 ⊢ (∀xφ → ∀y∀xφ)
3		ax-17 1616	. 2 ⊢ (¬ φ → ∀y ¬ φ)
4		biidd 228	. 2 ⊢ (x = y → (φ ↔ φ))
5	1, 2, 3, 4	spfw 1691	1 ⊢ (∀xφ → φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀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)