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Metamath Proof Explorer

Theorem cbval2vw

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypothesis cbval2vw.1 ⊢ x = z ∧ y = w → φ ↔ ψ
Assertion cbval2vw ⊢ ∀ x ∀ y φ ↔ ∀ z ∀ w ψ

Proof

Step Hyp Ref Expression
1 cbval2vw.1 ⊢ x = z ∧ y = w → φ ↔ ψ
2 1 cbvaldvaw ⊢ x = z → ∀ y φ ↔ ∀ w ψ
3 2 cbvalvw ⊢ ∀ x ∀ y φ ↔ ∀ z ∀ w ψ