Theorem brfvid 43117

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	⊢ (𝜑 → 𝑅 ∈ V)

Assertion

Ref	Expression
brfvid	⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Proof of Theorem brfvid

Step	Hyp	Ref	Expression
1		brfvid.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
2		fvi 6974	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
4	3	breqd 5159	1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  Vcvv 3471   class class class wbr 5148   I cid 5575  ‘cfv 6548

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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556

This theorem is referenced by: (None)