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Metamath Proof Explorer

Theorem cbv3

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbv3v if possible. (Contributed by NM, 5-Aug-1993) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbv3.1	⊢ Ⅎ 𝑦 𝜑
	cbv3.2	⊢ Ⅎ 𝑥 𝜓
	cbv3.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	cbv3	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbv3.1	⊢ Ⅎ 𝑦 𝜑
2		cbv3.2	⊢ Ⅎ 𝑥 𝜓
3		cbv3.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
5	4	hbal	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
6	2 3	spim	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
7	5 6	alrimih	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )