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Theorem brfvid 43678
Description: If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
Hypothesis
Ref Expression
brfvid.r (𝜑𝑅 ∈ V)
Assertion
Ref Expression
brfvid (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Proof of Theorem brfvid
StepHypRef Expression
1 brfvid.r . . 3 (𝜑𝑅 ∈ V)
2 fvi 6983 . . 3 (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
31, 2syl 17 . 2 (𝜑 → ( I ‘𝑅) = 𝑅)
43breqd 5152 1 (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3479   class class class wbr 5141   I cid 5575  cfv 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567
This theorem is referenced by: (None)
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