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Theorem bdinex1g 9332
Description: Bounded version of inex1g 3884. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdinex1g.bd BOUNDED B
Assertion
Ref Expression
bdinex1g (A 𝑉 → (AB) V)

Proof of Theorem bdinex1g
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ineq1 3125 . . 3 (x = A → (xB) = (AB))
21eleq1d 2103 . 2 (x = A → ((xB) V ↔ (AB) V))
3 bdinex1g.bd . . 3 BOUNDED B
4 vex 2554 . . 3 x V
53, 4bdinex1 9330 . 2 (xB) V
62, 5vtoclg 2607 1 (A 𝑉 → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  BOUNDED wbdc 9275
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  ax-bdsep 9319
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-v 2553  df-in 2918  df-bdc 9276
This theorem is referenced by: (None)
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