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Metamath Proof Explorer

Theorem rabbid

Description: Version of rabbidv with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

Ref Expression
Hypotheses rabbid.n ⊢ Ⅎ x φ
rabbid.1 ⊢ φ → ψ ↔ χ
Assertion rabbid ⊢ φ → x ∈ A | ψ = x ∈ A | χ

Proof

Step Hyp Ref Expression
1 rabbid.n ⊢ Ⅎ x φ
2 rabbid.1 ⊢ φ → ψ ↔ χ
3 2 adantr ⊢ φ ∧ x ∈ A → ψ ↔ χ
4 1 3 rabbida ⊢ φ → x ∈ A | ψ = x ∈ A | χ