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Metamath Proof Explorer

Theorem cbval

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out cbvalw , cbvalvw , cbvalv1 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993) (Revised by Mario Carneiro, 3-Oct-2016) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbval.1	⊢ Ⅎ 𝑦 𝜑
2		cbval.2	⊢ Ⅎ 𝑥 𝜓
3		cbval.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	3	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
5	1 2 4	cbv3	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )
6	3	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
7	6	equcoms	⊢ ( 𝑦 = 𝑥 → ( 𝜓 → 𝜑 ) )
8	2 1 7	cbv3	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜑 )
9	5 8	impbii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )