ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nffr GIF version

Theorem nffr 4239
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 4222 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
4 nfcv 2256 . . . 4 𝑥𝑠
52, 3, 4nffrfor 4238 . . 3 𝑥 FrFor 𝑅𝐴𝑠
65nfal 1538 . 2 𝑥𝑠 FrFor 𝑅𝐴𝑠
71, 6nfxfr 1433 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff set class
Syntax hints:  wal 1312  wnf 1419  wnfc 2243   FrFor wfrfor 4217   Fr wfr 4218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-frfor 4221  df-frind 4222
This theorem is referenced by:  nfwe  4245
  Copyright terms: Public domain W3C validator