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Theorem nffr 5525
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 5509 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏))
2 nfcv 2904 . . . . . 6 𝑥𝑎
3 nffr.a . . . . . 6 𝑥𝐴
42, 3nfss 3892 . . . . 5 𝑥 𝑎𝐴
5 nfv 1922 . . . . 5 𝑥 𝑎 ≠ ∅
64, 5nfan 1907 . . . 4 𝑥(𝑎𝐴𝑎 ≠ ∅)
7 nfcv 2904 . . . . . . . 8 𝑥𝑐
8 nffr.r . . . . . . . 8 𝑥𝑅
9 nfcv 2904 . . . . . . . 8 𝑥𝑏
107, 8, 9nfbr 5100 . . . . . . 7 𝑥 𝑐𝑅𝑏
1110nfn 1865 . . . . . 6 𝑥 ¬ 𝑐𝑅𝑏
122, 11nfralw 3147 . . . . 5 𝑥𝑐𝑎 ¬ 𝑐𝑅𝑏
132, 12nfrex 3228 . . . 4 𝑥𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏
146, 13nfim 1904 . . 3 𝑥((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
1514nfal 2322 . 2 𝑥𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
161, 15nfxfr 1860 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1541  wnf 1791  wnfc 2884  wne 2940  wral 3061  wrex 3062  wss 3866  c0 4237   class class class wbr 5053   Fr wfr 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-fr 5509
This theorem is referenced by:  nfwe  5527
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