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Theorem 3sstr4d 3201
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4d.1 (𝜑𝐴𝐵)
3sstr4d.2 (𝜑𝐶 = 𝐴)
3sstr4d.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3sstr4d (𝜑𝐶𝐷)

Proof of Theorem 3sstr4d
StepHypRef Expression
1 3sstr4d.1 . 2 (𝜑𝐴𝐵)
2 3sstr4d.2 . . 3 (𝜑𝐶 = 𝐴)
3 3sstr4d.3 . . 3 (𝜑𝐷 = 𝐵)
42, 3sseq12d 3187 . 2 (𝜑 → (𝐶𝐷𝐴𝐵))
51, 4mpbird 167 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wss 3130
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3136  df-ss 3143
This theorem is referenced by:  rdgss  6384  sucinc2  6447  oawordi  6470  nnnninf  7124  fzoss1  10171  fzoss2  10172  clsss  13621  ntrss  13622  sslm  13750  txss12  13769  metss2lem  14000  xmettxlem  14012  xmettx  14013  nnsf  14757  nninfself  14765
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