Theorem 3sstr4i 3289

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 3276	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 146	1 ⊢ 𝐶 ⊆ 𝐷

This proof depends on syntax axioms:   = wceq 1402   ⊆ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is used by:  undif2ss  3603  pwsnss  3927  iinuniss  4093  brab2a  4826  relopabiv  4901  rncoss  5051  imassrn  5135  rnin  5195  inimass  5202  imadiflem  5458  imainlem  5460  ssoprab2i  6171  npsspw  7832  axresscn  8221  mpomulf  8310  birthdaylem1g  16070