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

Metamath Proof Explorer

Theorem sb6rfv

Description: Reversed substitution. Version of sb6rf requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993) (Revised by Wolf Lammen, 7-Feb-2023)

		Ref	Expression
	Hypothesis	sb6rfv.nf	⊢ Ⅎ 𝑦 𝜑
	Assertion	sb6rfv	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb6rfv.nf	⊢ Ⅎ 𝑦 𝜑
2		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	equsalv	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ 𝜑 )
4	3	bicomi	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )