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Theorem 3sstr4i 3065
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 𝐴𝐵
3sstr4.2 𝐶 = 𝐴
3sstr4.3 𝐷 = 𝐵
Assertion
Ref Expression
3sstr4i 𝐶𝐷

Proof of Theorem 3sstr4i
StepHypRef Expression
1 3sstr4.1 . 2 𝐴𝐵
2 3sstr4.2 . . 3 𝐶 = 𝐴
3 3sstr4.3 . . 3 𝐷 = 𝐵
42, 3sseq12i 3052 . 2 (𝐶𝐷𝐴𝐵)
51, 4mpbir 144 1 𝐶𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012
This theorem is referenced by:  undif2ss  3358  pwsnss  3647  iinuniss  3811  brab2a  4491  rncoss  4703  imassrn  4785  rnin  4841  inimass  4848  imadiflem  5093  imainlem  5095  ssoprab2i  5737  npsspw  7030  axresscn  7397
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