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Metamath Proof Explorer

Theorem eqsstrrd

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

Ref Expression
Hypotheses eqsstrrd.1 ⊢ ( 𝜑 → 𝐵 = 𝐴 )
eqsstrrd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion eqsstrrd ( 𝜑 → 𝐴 ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 eqsstrrd.1 ⊢ ( 𝜑 → 𝐵 = 𝐴 )
2 eqsstrrd.2 ⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
3 1 eqcomd ⊢ ( 𝜑 → 𝐴 = 𝐵 )
4 3 2 eqsstrd ⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )