Theorem 3sstr3i 4019

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 4007	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	mpbi 229	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1533   ⊆ wss 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-ss 3961

This theorem is referenced by:  ttrclco  9743  cottrcl  9744  odf1o2  19540  leordtval2  23160  uniiccvol  25553  ballotlem2  34239  cotrcltrcl  43297