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Theorem 3sstr4d 2961
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 2947 . 2 (φ → (𝐶𝐷AB))
51, 4mpbird 156 1 (φ𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226  wss 2890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904
This theorem is referenced by:  rdgss  5886  sucinc2  5937  oawordi  5960
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