Theorem 3sstr3i 4051

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 4039	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	mpbi 230	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1537   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-ss 3993

This theorem is referenced by:  ttrclco  9787  cottrcl  9788  odf1o2  19615  leordtval2  23241  uniiccvol  25634  ballotlem2  34453  cotrcltrcl  43687