Theorem 3sstr4i 3311

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
3sstr4.1	⊢ A ⊆ B
3sstr4.2	⊢ C = A
3sstr4.3	⊢ D = B

Assertion

Ref	Expression
3sstr4i	⊢ C ⊆ D

Proof of Theorem 3sstr4i

Step	Hyp	Ref	Expression
1		3sstr4.1	. 2 ⊢ A ⊆ B
2		3sstr4.2	. . 3 ⊢ C = A
3		3sstr4.3	. . 3 ⊢ D = B
4	2, 3	sseq12i 3298	. 2 ⊢ (C ⊆ D ↔ A ⊆ B)
5	1, 4	mpbir 200	1 ⊢ C ⊆ D

Syntax hints:   = wceq 1642   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  rncoss  4973  imassrn  5010  rnin  5038  ssoprab2i  5581