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Theorem brfvid 38500
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 6419 . . 3 (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
31, 2syl 17 . 2 (𝜑 → ( I ‘𝑅) = 𝑅)
43breqd 4816 1 (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2140  Vcvv 3341   class class class wbr 4805   I cid 5174  cfv 6050
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058
This theorem is referenced by: (None)
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