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Theorem eqsstrrd 3134
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
eqsstrrd.1 (𝜑𝐵 = 𝐴)
eqsstrrd.2 (𝜑𝐵𝐶)
Assertion
Ref Expression
eqsstrrd (𝜑𝐴𝐶)

Proof of Theorem eqsstrrd
StepHypRef Expression
1 eqsstrrd.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2145 . 2 (𝜑𝐴 = 𝐵)
3 eqsstrrd.2 . 2 (𝜑𝐵𝐶)
42, 3eqsstrd 3133 1 (𝜑𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  ssxpbm  4974  ssxp1  4975  ssxp2  4976  suppssof1  5999  tfrlemiubacc  6227  tfr1onlemubacc  6243  tfrcllemubacc  6256  oaword1  6367  phplem4dom  6756  fisseneq  6820  archnqq  7232  epttop  12269  metequiv2  12675  limccnpcntop  12823  limccnp2lem  12824  limccnp2cntop  12825  nnsf  13213
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