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Metamath Proof Explorer

Theorem 3sstr3d

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000)

Ref Expression
Hypotheses 3sstr3d.1 ⊢ φ → A ⊆ B
3sstr3d.2 ⊢ φ → A = C
3sstr3d.3 ⊢ φ → B = D
Assertion 3sstr3d ⊢ φ → C ⊆ D

Proof

Step Hyp Ref Expression
1 3sstr3d.1 ⊢ φ → A ⊆ B
2 3sstr3d.2 ⊢ φ → A = C
3 3sstr3d.3 ⊢ φ → B = D
4 2 1 eqsstrrd ⊢ φ → C ⊆ B
5 4 3 sseqtrd ⊢ φ → C ⊆ D