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Theorem dffr2ALT 5642
Description: Alternate proof of dffr2 5641, which avoids ax-8 2109 but requires ax-10 2138, ax-11 2155, ax-12 2172. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dffr2ALT (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧

Proof of Theorem dffr2ALT
StepHypRef Expression
1 df-fr 5632 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
2 rabeq0 4385 . . . . 5 ({𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∀𝑧𝑥 ¬ 𝑧𝑅𝑦)
32rexbii 3095 . . . 4 (∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅ ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦)
43imbi2i 336 . . 3 (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
54albii 1822 . 2 (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
61, 5bitr4i 278 1 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 {𝑧𝑥𝑧𝑅𝑦} = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wne 2941  wral 3062  wrex 3071  {crab 3433  wss 3949  c0 4323   class class class wbr 5149   Fr wfr 5629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-ral 3063  df-rex 3072  df-rab 3434  df-dif 3952  df-nul 4324  df-fr 5632
This theorem is referenced by: (None)
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