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Metamath Proof Explorer

Theorem cbval2vv

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbval2vw if possible. (Contributed by NM, 4-Feb-2005) Remove dependency on ax-10 . (Revised by Wolf Lammen, 18-Jul-2021) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	cbval2vv.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
	Assertion	cbval2vv	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbval2vv.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
2	1	cbvaldva	⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 𝜑 ↔ ∀ 𝑤 𝜓 ) )
3	2	cbvalv	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 𝜓 )