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Metamath Proof Explorer

Theorem sbcbi

Description: Implication form of sbcbii . sbcbi is sbcbiVD without virtual deductions and was automatically derived from sbcbiVD using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sbcbi ⊢ A ∈ V → ∀ x φ ↔ ψ → [˙A / x]˙ φ ↔ [˙A / x]˙ ψ

Proof

Step Hyp Ref Expression
1 spsbc ⊢ A ∈ V → ∀ x φ ↔ ψ → [˙A / x]˙ φ ↔ ψ
2 sbcbig ⊢ A ∈ V → [˙A / x]˙ φ ↔ ψ ↔ [˙A / x]˙ φ ↔ [˙A / x]˙ ψ
3 1 2 sylibd ⊢ A ∈ V → ∀ x φ ↔ ψ → [˙A / x]˙ φ ↔ [˙A / x]˙ ψ