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Metamath Proof Explorer

Theorem cbvex2vv

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 cbvex2vw if possible. (Contributed by NM, 26-Jul-1995) 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 cbvex2vv ⊢ ∃ x ∃ y φ ↔ ∃ z ∃ w ψ

Proof

Step Hyp Ref Expression
1 cbval2vv.1 ⊢ x = z ∧ y = w → φ ↔ ψ
2 1 cbvexdva ⊢ x = z → ∃ y φ ↔ ∃ w ψ
3 2 cbvexv ⊢ ∃ x ∃ y φ ↔ ∃ z ∃ w ψ