Theorem sb6rf

Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb6rfv if possible. (Contributed by NM, 1-Aug-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 21-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb5rf.1	⊢ Ⅎ 𝑦 𝜑
Assertion	sb6rf	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		sb5rf.1	⊢ Ⅎ 𝑦 𝜑
2		sbequ12r	⊢ ( 𝑦 = 𝑥 → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜑 ) )
3	1 2	equsal	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ 𝜑 )
4	3	bicomi	⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )