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Theorem coeq2 4769
Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem coeq2
StepHypRef Expression
1 coss2 4767 . . 3 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
2 coss2 4767 . . 3 (𝐵𝐴 → (𝐶𝐵) ⊆ (𝐶𝐴))
31, 2anim12i 336 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐶𝐴) ⊆ (𝐶𝐵) ∧ (𝐶𝐵) ⊆ (𝐶𝐴)))
4 eqss 3162 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3162 . 2 ((𝐶𝐴) = (𝐶𝐵) ↔ ((𝐶𝐴) ⊆ (𝐶𝐵) ∧ (𝐶𝐵) ⊆ (𝐶𝐴)))
63, 4, 53imtr4i 200 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wss 3121  ccom 4615
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-br 3990  df-opab 4051  df-co 4620
This theorem is referenced by:  coeq2i  4771  coeq2d  4773  coi2  5127  relcnvtr  5130  relcoi1  5142  f1eqcocnv  5770  ereq1  6520  upxp  13066  uptx  13068  txcn  13069
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