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Metamath Proof Explorer

Theorem sbcex

Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016)

Ref Expression
Assertion sbcex ⊢ [˙A / x]˙ φ → A ∈ V

Proof

Step Hyp Ref Expression
1 df-sbc ⊢ [˙A / x]˙ φ ↔ A ∈ x | φ
2 elex ⊢ A ∈ x | φ → A ∈ V
3 1 2 sylbi ⊢ [˙A / x]˙ φ → A ∈ V