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Theorem cleq1lem 14432
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 3903 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21anbi1d 633 1 (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wss 3844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-in 3851  df-ss 3861
This theorem is referenced by:  cleq1  14433  trcleq12lem  14443  lcmfun  16087  coprmproddvds  16105  isslw  18852  neival  21854  nrmsep3  22107  xkococnlem  22411  ovolval  24226  ovnval2b  43624
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