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Theorem cleq1lem 14322
Description: Equality implies bijection. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
cleq1lem (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))

Proof of Theorem cleq1lem
StepHypRef Expression
1 sseq1 3971 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21anbi1d 631 1 (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wss 3913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475  df-in 3920  df-ss 3930
This theorem is referenced by:  cleq1  14323  trcleq12lem  14333  lcmfun  15967  coprmproddvds  15985  isslw  18712  neival  21686  nrmsep3  21939  xkococnlem  22243  ovolval  24056  ovnval2b  42982
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