Theorem 3sstr3i 3959

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 3956	. 2 ⊢ 𝐶 ⊆ 𝐵
4		3sstr3.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	sseqtri 3957	1 ⊢ 𝐶 ⊆ 𝐷

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3897

This theorem is used by:  ttrclco  9726  cottrcl  9727  odf1o2  19796  leordtval2  23539  uniiccvol  25910  ballotlem2  35171  cotrcltrcl  44862