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Theorem sseq2 2961
Description: Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
Assertion
Ref Expression
sseq2 (A = B → (𝐶A𝐶B))

Proof of Theorem sseq2
StepHypRef Expression
1 sstr2 2946 . . . 4 (𝐶A → (AB𝐶B))
21com12 27 . . 3 (AB → (𝐶A𝐶B))
3 sstr2 2946 . . . 4 (𝐶B → (BA𝐶A))
43com12 27 . . 3 (BA → (𝐶B𝐶A))
52, 4anim12i 321 . 2 ((AB BA) → ((𝐶A𝐶B) (𝐶B𝐶A)))
6 eqss 2954 . 2 (A = B ↔ (AB BA))
7 dfbi2 368 . 2 ((𝐶A𝐶B) ↔ ((𝐶A𝐶B) (𝐶B𝐶A)))
85, 6, 73imtr4i 190 1 (A = B → (𝐶A𝐶B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wss 2911
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925
This theorem is referenced by:  sseq12  2962  sseq2i  2964  sseq2d  2967  syl5sseq  2987  nssne1  2995  psseq2  3026  sseq0  3252  un00  3257  disjpss  3272  pweq  3354  ssintab  3623  ssintub  3624  intmin  3626  treq  3851  ssexg  3887  iunpw  4177  ordtri2orexmid  4211  onsucsssucexmid  4212  fununi  4910  funcnvuni  4911  feq3  4975  ssimaexg  5178  nnawordex  6037  ereq1  6049  xpiderm  6113  domeng  6169  ssfiexmid  6254  bdssexg  9359  bj-nntrans  9411  bj-omtrans  9416
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