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Theorem nffr 4113
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 variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-frind 4096 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
4 nfcv 2194 . . . 4 𝑥𝑠
52, 3, 4nffrfor 4112 . . 3 𝑥 FrFor 𝑅𝐴𝑠
65nfal 1484 . 2 𝑥𝑠 FrFor 𝑅𝐴𝑠
71, 6nfxfr 1379 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff set class
Syntax hints:  wal 1257  wnf 1365  wnfc 2181   FrFor wfrfor 4091   Fr wfr 4092
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
This theorem is referenced by:  nfwe  4119
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