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Theorem nfwe 4119
Description: Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfwe.r 𝑥𝑅
nfwe.a 𝑥𝐴
Assertion
Ref Expression
nfwe 𝑥 𝑅 We 𝐴

Proof of Theorem nfwe
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wetr 4098 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfwe.r . . . 4 𝑥𝑅
3 nfwe.a . . . 4 𝑥𝐴
42, 3nffr 4113 . . 3 𝑥 𝑅 Fr 𝐴
5 nfcv 2194 . . . . . . . . 9 𝑥𝑎
6 nfcv 2194 . . . . . . . . 9 𝑥𝑏
75, 2, 6nfbr 3835 . . . . . . . 8 𝑥 𝑎𝑅𝑏
8 nfcv 2194 . . . . . . . . 9 𝑥𝑐
96, 2, 8nfbr 3835 . . . . . . . 8 𝑥 𝑏𝑅𝑐
107, 9nfan 1473 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
115, 2, 8nfbr 3835 . . . . . . 7 𝑥 𝑎𝑅𝑐
1210, 11nfim 1480 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
133, 12nfralxy 2377 . . . . 5 𝑥𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
143, 13nfralxy 2377 . . . 4 𝑥𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
153, 14nfralxy 2377 . . 3 𝑥𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
164, 15nfan 1473 . 2 𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
171, 16nfxfr 1379 1 𝑥 𝑅 We 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wnf 1365  wnfc 2181  wral 2323   class class class wbr 3791   Fr wfr 4092   We wwe 4094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-frfor 4095  df-frind 4096  df-wetr 4098
This theorem is referenced by: (None)
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