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Theorem bj-indsuc 9317
Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indsuc (Ind A → (B A → suc B A))

Proof of Theorem bj-indsuc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 9316 . . 3 (Ind A ↔ (∅ A x A suc x A))
21simprbi 260 . 2 (Ind Ax A suc x A)
3 suceq 4105 . . . 4 (x = B → suc x = suc B)
43eleq1d 2103 . . 3 (x = B → (suc x A ↔ suc B A))
54rspcv 2646 . 2 (B A → (x A suc x A → suc B A))
62, 5syl5com 26 1 (Ind A → (B A → suc B A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wral 2300  c0 3218  suc csuc 4068  Ind wind 9315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-suc 4074  df-bj-ind 9316
This theorem is referenced by:  bj-indint  9320  bj-peano2  9327  bj-inf2vnlem2  9355
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