Theorem 3sstr4i 3233

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 3220	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 146	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1372   ⊆ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  undif2ss  3535  pwsnss  3843  iinuniss  4009  brab2a  4726  relopabiv  4799  rncoss  4946  imassrn  5030  rnin  5089  inimass  5096  imadiflem  5347  imainlem  5349  ssoprab2i  6024  npsspw  7566  axresscn  7955  mpomulf  8044