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Theorem coeq1 4520
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 4518 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 coss1 4518 . . 3 (𝐵𝐴 → (𝐵𝐶) ⊆ (𝐴𝐶))
31, 2anim12i 325 . 2 ((𝐴𝐵𝐵𝐴) → ((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ⊆ (𝐴𝐶)))
4 eqss 2987 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 2987 . 2 ((𝐴𝐶) = (𝐵𝐶) ↔ ((𝐴𝐶) ⊆ (𝐵𝐶) ∧ (𝐵𝐶) ⊆ (𝐴𝐶)))
63, 4, 53imtr4i 194 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wss 2944  ccom 4376
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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  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-nfc 2183  df-in 2951  df-ss 2958  df-br 3792  df-opab 3846  df-co 4381
This theorem is referenced by:  coeq1i  4522  coeq1d  4524  coi2  4864  relcnvtr  4867  funcoeqres  5184  ereq1  6143
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