This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb9i

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

		Ref	Expression
	Assertion	sb9i	⊢ ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb9	⊢ ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
2	1	biimpi	⊢ ( ∀ 𝑥 [ 𝑥 / 𝑦 ] 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )