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Theorem bdinex1 6801
 Description: Bounded version of inex1 3863. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdinex1.bd BOUNDED B
bdinex1.1 A V
Assertion
Ref Expression
bdinex1 (AB) V

Proof of Theorem bdinex1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdinex1.1 . . . 4 A V
2 bdinex1.bd . . . . . 6 BOUNDED B
32bdeli 6752 . . . . 5 BOUNDED y B
43bdzfauscl 6795 . . . 4 (A V → xy(y x ↔ (y A y B)))
51, 4ax-mp 7 . . 3 xy(y x ↔ (y A y B))
6 dfcleq 2016 . . . . 5 (x = (AB) ↔ y(y xy (AB)))
7 elin 3101 . . . . . . 7 (y (AB) ↔ (y A y B))
87bibi2i 216 . . . . . 6 ((y xy (AB)) ↔ (y x ↔ (y A y B)))
98albii 1339 . . . . 5 (y(y xy (AB)) ↔ y(y x ↔ (y A y B)))
106, 9bitri 173 . . . 4 (x = (AB) ↔ y(y x ↔ (y A y B)))
1110exbii 1478 . . 3 (x x = (AB) ↔ xy(y x ↔ (y A y B)))
125, 11mpbir 134 . 2 x x = (AB)
1312issetri 2540 1 (AB) V
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2533   ∩ cin 2891  BOUNDED wbdc 6746 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bdsep 6790 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2535  df-in 2899  df-bdc 6747 This theorem is referenced by:  bdinex2  6802  bdinex1g  6803  bdpeano5  6844
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