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Theorem 3sstr4i 3623
Description: Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
3sstr4.1 𝐴𝐵
3sstr4.2 𝐶 = 𝐴
3sstr4.3 𝐷 = 𝐵
Assertion
Ref Expression
3sstr4i 𝐶𝐷

Proof of Theorem 3sstr4i
StepHypRef Expression
1 3sstr4.1 . 2 𝐴𝐵
2 3sstr4.2 . . 3 𝐶 = 𝐴
3 3sstr4.3 . . 3 𝐷 = 𝐵
42, 3sseq12i 3610 . 2 (𝐶𝐷𝐴𝐵)
51, 4mpbir 221 1 𝐶𝐷
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3562  df-ss 3569
This theorem is referenced by:  brab2a  5129  rncoss  5346  imassrn  5436  rnin  5501  inimass  5508  ssoprab2i  6702  omopthlem2  7681  rankval4  8674  cardf2  8713  r0weon  8779  dcomex  9213  axdc2lem  9214  fpwwe2lem1  9397  canthwe  9417  recmulnq  9730  npex  9752  axresscn  9913  trclublem  13668  bpoly4  14715  2strop1  15909  odlem1  17875  gexlem1  17915  psrbagsn  19414  bwth  21123  2ndcctbss  21168  uniioombllem4  23260  uniioombllem5  23261  eff1olem  24198  birthdaylem1  24578  nvss  27294  lediri  28242  lejdiri  28244  sshhococi  28251  mayetes3i  28434  disjxpin  29243  imadifxp  29256  sxbrsigalem5  30128  eulerpartlemmf  30215  kur14lem6  30898  cvmlift2lem12  31001  bj-rrhatsscchat  32753  mblfinlem4  33078  lclkrs2  36306  areaquad  37280  corclrcl  37477  corcltrcl  37509  relopabVD  38617
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