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Metamath Proof Explorer

Theorem sb8

Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 . For a version requiring disjoint variables, but fewer axioms, see sb8f . (Contributed by NM, 16-May-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Jim Kingdon, 15-Jan-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb8.1	⊢ Ⅎ 𝑦 𝜑
	Assertion	sb8	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb8.1	⊢ Ⅎ 𝑦 𝜑
2	1	nfs1	⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3		sbequ12	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
4	1 2 3	cbval	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )