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Mirrors > Home > ILE Home > Th. List > ssrdv | GIF version |
Description: Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.) |
Ref | Expression |
---|---|
ssrdv.1 | ⊢ (φ → (x ∈ A → x ∈ B)) |
Ref | Expression |
---|---|
ssrdv | ⊢ (φ → A ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrdv.1 | . . 3 ⊢ (φ → (x ∈ A → x ∈ B)) | |
2 | 1 | alrimiv 1751 | . 2 ⊢ (φ → ∀x(x ∈ A → x ∈ B)) |
3 | dfss2 2928 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
4 | 2, 3 | sylibr 137 | 1 ⊢ (φ → A ⊆ B) |
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