Theorem sbalv

Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypothesis	sbalv.1	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
Assertion	sbalv	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbalv.1	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
2		sbal	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑧 𝜑 ↔ ∀ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
3	1	albii	⊢ ( ∀ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑧 𝜓 )
4	2 3	bitri	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑧 𝜑 ↔ ∀ 𝑧 𝜓 )