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Theorem dffr6 5640
Description: Alternate definition of df-fr 5637. See dffr5 35754 for a definition without dummy variables (but note that their equivalence uses ax-sep 5296). (Contributed by BJ, 16-Nov-2024.)
Assertion
Ref Expression
dffr6 (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr6
StepHypRef Expression
1 velpw 4605 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
21bicomi 224 . . . . . 6 (𝑥𝐴𝑥 ∈ 𝒫 𝐴)
3 velsn 4642 . . . . . . . 8 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43bicomi 224 . . . . . . 7 (𝑥 = ∅ ↔ 𝑥 ∈ {∅})
54necon3abii 2987 . . . . . 6 (𝑥 ≠ ∅ ↔ ¬ 𝑥 ∈ {∅})
62, 5anbi12i 628 . . . . 5 ((𝑥𝐴𝑥 ≠ ∅) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
7 eldif 3961 . . . . 5 (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ¬ 𝑥 ∈ {∅}))
86, 7bitr4i 278 . . . 4 ((𝑥𝐴𝑥 ≠ ∅) ↔ 𝑥 ∈ (𝒫 𝐴 ∖ {∅}))
98imbi1i 349 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ (𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
109albii 1819 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
11 df-fr 5637 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
12 df-ral 3062 . 2 (∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∀𝑥(𝑥 ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
1310, 11, 123bitr4i 303 1 (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143   Fr wfr 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-dif 3954  df-ss 3968  df-pw 4602  df-sn 4627  df-fr 5637
This theorem is referenced by:  frd  5641
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