Theorem 3sstr3i 3995

Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr3.1	⊢ 𝐴 ⊆ 𝐵
3sstr3.2	⊢ 𝐴 = 𝐶
3sstr3.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3sstr3i	⊢ 𝐶 ⊆ 𝐷

Proof of Theorem 3sstr3i

Step	Hyp	Ref	Expression
1		3sstr3.2	. . 3 ⊢ 𝐴 = 𝐶
2		3sstr3.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
3	1, 2	eqsstrri 3992	. 2 ⊢ 𝐶 ⊆ 𝐵
4		3sstr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	sseqtri 3993	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1567   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ss 3930

This theorem is referenced by:  ttrclco  9686  cottrcl  9687  odf1o2  19642  leordtval2  23337  uniiccvol  25707  ballotlem2  34823  cotrcltrcl  44342