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Theorem friOLD 5643
Description: Obsolete version of fri 5642 as of 16-Nov-2024. (Contributed by NM, 18-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
friOLD (((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem friOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-fr 5637 . . 3 (𝑅 Fr 𝐴 ↔ ∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥))
2 sseq1 4009 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝐴𝐵𝐴))
3 neeq1 3003 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ≠ ∅ ↔ 𝐵 ≠ ∅))
42, 3anbi12d 632 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝐴𝑧 ≠ ∅) ↔ (𝐵𝐴𝐵 ≠ ∅)))
5 raleq 3323 . . . . . 6 (𝑧 = 𝐵 → (∀𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐵 ¬ 𝑦𝑅𝑥))
65rexeqbi1dv 3339 . . . . 5 (𝑧 = 𝐵 → (∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥 ↔ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥))
74, 6imbi12d 344 . . . 4 (𝑧 = 𝐵 → (((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) ↔ ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
87spcgv 3596 . . 3 (𝐵𝐶 → (∀𝑧((𝑧𝐴𝑧 ≠ ∅) → ∃𝑥𝑧𝑦𝑧 ¬ 𝑦𝑅𝑥) → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
91, 8biimtrid 242 . 2 (𝐵𝐶 → (𝑅 Fr 𝐴 → ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)))
109imp31 417 1 (((𝐵𝐶𝑅 Fr 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1538   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333   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-rex 3071  df-v 3482  df-ss 3968  df-fr 5637
This theorem is referenced by: (None)
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