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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 )