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Theorem cleq1lem 14876
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 3973 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21anbi1d 631 1 (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wss 3914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931
This theorem is referenced by:  cleq1  14877  trcleq12lem  14887  lcmfun  16529  coprmproddvds  16547  isslw  19398  neival  22476  nrmsep3  22729  xkococnlem  23033  ovolval  24860  ovnval2b  44883
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