Theorem 3sstr4i 3268

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 3255	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 146	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1397   ⊆ wss 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  undif2ss  3570  pwsnss  3887  iinuniss  4053  brab2a  4779  relopabiv  4853  rncoss  5003  imassrn  5087  rnin  5146  inimass  5153  imadiflem  5409  imainlem  5411  ssoprab2i  6109  npsspw  7690  axresscn  8079  mpomulf  8168