Metamath Proof Explorer

Theorem bj-ceqsaltv

Description: Version of bj-ceqsalt with a disjoint variable condition on x , V , removing dependency on df-sb and df-clab . Prefer its use over bj-ceqsalt when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

Ref Expression
Assertion bj-ceqsaltv ⊢ Ⅎ x ψ ∧ ∀ x x = A → φ ↔ ψ ∧ A ∈ V → ∀ x x = A → φ ↔ ψ

Proof

Step Hyp Ref Expression
1 elissetv ⊢ A ∈ V → ∃ x x = A
2 1 3anim3i ⊢ Ⅎ x ψ ∧ ∀ x x = A → φ ↔ ψ ∧ A ∈ V → Ⅎ x ψ ∧ ∀ x x = A → φ ↔ ψ ∧ ∃ x x = A
3 bj-ceqsalt0 ⊢ Ⅎ x ψ ∧ ∀ x x = A → φ ↔ ψ ∧ ∃ x x = A → ∀ x x = A → φ ↔ ψ
4 2 3 syl ⊢ Ⅎ x ψ ∧ ∀ x x = A → φ ↔ ψ ∧ A ∈ V → ∀ x x = A → φ ↔ ψ