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Metamath Proof Explorer

Theorem cbvalv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See cbvalvw for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993) Remove dependency on ax-10 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbvalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	cbvalv	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbval	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )