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Metamath Proof Explorer

Theorem sseqtrd

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004)

Ref Expression
Hypotheses sseqtrd.1 ⊢ φ → A ⊆ B
sseqtrd.2 ⊢ φ → B = C
Assertion sseqtrd ⊢ φ → A ⊆ C

Proof

Step Hyp Ref Expression
1 sseqtrd.1 ⊢ φ → A ⊆ B
2 sseqtrd.2 ⊢ φ → B = C
3 2 sseq2d ⊢ φ → A ⊆ B ↔ A ⊆ C
4 1 3 mpbid ⊢ φ → A ⊆ C