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Theorem nffr 5012
 Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nffr 𝑥 𝑅 Fr 𝐴

Proof of Theorem nffr
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 4997 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏))
2 nfcv 2751 . . . . . 6 𝑥𝑎
3 nffr.a . . . . . 6 𝑥𝐴
42, 3nfss 3561 . . . . 5 𝑥 𝑎𝐴
5 nfv 1830 . . . . 5 𝑥 𝑎 ≠ ∅
64, 5nfan 1816 . . . 4 𝑥(𝑎𝐴𝑎 ≠ ∅)
7 nfcv 2751 . . . . . . . 8 𝑥𝑐
8 nffr.r . . . . . . . 8 𝑥𝑅
9 nfcv 2751 . . . . . . . 8 𝑥𝑏
107, 8, 9nfbr 4629 . . . . . . 7 𝑥 𝑐𝑅𝑏
1110nfn 1768 . . . . . 6 𝑥 ¬ 𝑐𝑅𝑏
122, 11nfral 2929 . . . . 5 𝑥𝑐𝑎 ¬ 𝑐𝑅𝑏
132, 12nfrex 2990 . . . 4 𝑥𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏
146, 13nfim 1813 . . 3 𝑥((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
1514nfal 2139 . 2 𝑥𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
161, 15nfxfr 1771 1 𝑥 𝑅 Fr 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699  Ⅎwnfc 2738   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-fr 4997 This theorem is referenced by:  nfwe  5014
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