Theorem 3sstr3i 4038

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		3sstr3.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		3sstr3.2	. . 3 ⊢ 𝐴 = 𝐶
3		3sstr3.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	sseq12i 4026	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	mpbi 230	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1537   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ss 3980

This theorem is referenced by:  ttrclco  9756  cottrcl  9757  odf1o2  19606  leordtval2  23236  uniiccvol  25629  ballotlem2  34470  cotrcltrcl  43715