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Theorem brfvid 43635
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 6979 . . 3 (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
31, 2syl 17 . 2 (𝜑 → ( I ‘𝑅) = 𝑅)
43breqd 5160 1 (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1535  wcel 2104  Vcvv 3477   class class class wbr 5149   I cid 5575  cfv 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6510  df-fun 6560  df-fv 6566
This theorem is referenced by: (None)
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