Theorem brfvid 43677

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 6965	. . 3 ⊢ (𝑅 ∈ V → ( I ‘𝑅) = 𝑅)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ( I ‘𝑅) = 𝑅)
4	3	breqd 5134	1 ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  Vcvv 3463   class class class wbr 5123   I cid 5557  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549

This theorem is referenced by: (None)