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Metamath Proof Explorer

Theorem cbvald

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . See cbvaldw for a version with x , y disjoint, not depending on ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvald.1	⊢ Ⅎ 𝑦 𝜑
	cbvald.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
	cbvald.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
Assertion	cbvald	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvald.1	⊢ Ⅎ 𝑦 𝜑
2		cbvald.2	⊢ ( 𝜑 → Ⅎ 𝑦 𝜓 )
3		cbvald.3	⊢ ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) ) )
4		nfv	⊢ Ⅎ 𝑥 𝜑
5		nfvd	⊢ ( 𝜑 → Ⅎ 𝑥 𝜒 )
6	4 1 2 5 3	cbv2	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )