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 ( 𝜑𝐴𝐵 )
3sstr3d.2 ( 𝜑𝐴 = 𝐶 )
3sstr3d.3 ( 𝜑𝐵 = 𝐷 )
Assertion 3sstr3d ( 𝜑𝐶𝐷 )

Proof

Step Hyp Ref Expression
1 3sstr3d.1 ( 𝜑𝐴𝐵 )
2 3sstr3d.2 ( 𝜑𝐴 = 𝐶 )
3 3sstr3d.3 ( 𝜑𝐵 = 𝐷 )
4 2 3 sseq12d ( 𝜑 → ( 𝐴𝐵𝐶𝐷 ) )
5 1 4 mpbid ( 𝜑𝐶𝐷 )