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Metamath Proof Explorer

Theorem 3sstr3g

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

Ref Expression
Hypotheses 3sstr3g.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
3sstr3g.2 ⊢ 𝐴 = 𝐶
3sstr3g.3 ⊢ 𝐵 = 𝐷
Assertion 3sstr3g ( 𝜑 → 𝐶 ⊆ 𝐷 )

Proof

Step Hyp Ref Expression
1 3sstr3g.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2 3sstr3g.2 ⊢ 𝐴 = 𝐶
3 3sstr3g.3 ⊢ 𝐵 = 𝐷
4 2 1 eqsstrrid ⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 )
5 4 3 sseqtrdi ⊢ ( 𝜑 → 𝐶 ⊆ 𝐷 )