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Metamath Proof Explorer

Theorem cbvaldva

Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbvaldvaw if possible. (Contributed by David Moews, 1-May-2017) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbvaldva.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3		nfvd	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
4	1	ex	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
5	2 3 4	cbvald	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )