Theorem 3sstr4i 3238

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 3225	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 146	1 ⊢ 𝐶 ⊆ 𝐷

Syntax hints:   = wceq 1373   ⊆ wss 3170

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3176  df-ss 3183

This theorem is referenced by:  undif2ss  3540  pwsnss  3850  iinuniss  4016  brab2a  4736  relopabiv  4809  rncoss  4958  imassrn  5042  rnin  5101  inimass  5108  imadiflem  5362  imainlem  5364  ssoprab2i  6047  npsspw  7604  axresscn  7993  mpomulf  8082