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Theorem brfvid 38296
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 6294 . . 3 (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
31, 2syl 17 . 2 (𝜑 → ( I ‘𝑅) = 𝑅)
43breqd 4696 1 (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685   I cid 5052  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by: (None)
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