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Theorem 3sstr4g 3014
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 2999 . 2 (𝐶𝐷𝐴𝐵)
51, 4sylibr 141 1 (𝜑𝐶𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959
This theorem is referenced by:  rabss2  3051  unss2  3142  sslin  3191  ssopab2  4040  xpss12  4473  coss1  4519  coss2  4520  cnvss  4536  rnss  4592  ssres  4665  ssres2  4666  imass1  4728  imass2  4729  imadif  5007  imain  5009  ssoprab2  5589  suppssfv  5736  suppssov1  5737  tposss  5892
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