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Metamath Proof Explorer

Theorem eqsstrrdi

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017)

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

Proof

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