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Theorem coeq1 4493
 Description: Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
Assertion
Ref Expression
coeq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem coeq1
StepHypRef Expression
1 coss1 4491 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 coss1 4491 . . 3 (𝐵𝐴 → (𝐵𝐶) ⊆ (𝐴𝐶))
31, 2anim12i 321 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ⊆ (𝐴𝐶)))
4 eqss 2960 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 2960 . 2 ((𝐴𝐶) = (𝐵𝐶) ↔ ((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ⊆ (𝐴𝐶)))
63, 4, 53imtr4i 190 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ⊆ wss 2917   ∘ ccom 4349 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931  df-br 3765  df-opab 3819  df-co 4354 This theorem is referenced by:  coeq1i  4495  coeq1d  4497  coi2  4837  relcnvtr  4840  funcoeqres  5157  ereq1  6113
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