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Theorem 3sstr4g 3140
Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4g.1 (𝜑𝐴𝐵)
3sstr4g.2 𝐶 = 𝐴
3sstr4g.3 𝐷 = 𝐵
Assertion
Ref Expression
3sstr4g (𝜑𝐶𝐷)

Proof of Theorem 3sstr4g
StepHypRef Expression
1 3sstr4g.1 . 2 (𝜑𝐴𝐵)
2 3sstr4g.2 . . 3 𝐶 = 𝐴
3 3sstr4g.3 . . 3 𝐷 = 𝐵
42, 3sseq12i 3125 . 2 (𝐶𝐷𝐴𝐵)
51, 4sylibr 133 1 (𝜑𝐶𝐷)
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:  rabss2  3180  unss2  3247  sslin  3302  ssopab2  4197  xpss12  4646  coss1  4694  coss2  4695  cnvss  4712  rnss  4769  ssres  4845  ssres2  4846  imass1  4914  imass2  4915  imadif  5203  imain  5205  ssoprab2  5827  suppssfv  5978  suppssov1  5979  tposss  6143  ss2ixp  6605  isumsplit  11272  isumrpcl  11275
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