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Theorem cleq1lem 13942
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 3767 . 2 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21anbi1d 743 1 (𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729
This theorem is referenced by:  cleq1  13943  trcleq12lem  13953  lcmfun  15580  coprmproddvds  15599  nrmsep3  21381  ovnval2b  41290
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