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Theorem cleq1lem 14928
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 4007 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21anbi1d 630 1 (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965
This theorem is referenced by:  cleq1  14929  trcleq12lem  14939  lcmfun  16581  coprmproddvds  16599  isslw  19475  neival  22605  nrmsep3  22858  xkococnlem  23162  ovolval  24989  ovnval2b  45258
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