Theorem bdinex2 14691

Description: Bounded version of inex2 4140. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdinex2.bd	|- BOUNDED B
bdinex2.1	|- A e. _V

Assertion

Ref	Expression
bdinex2	|- ( B i^i A ) e. _V

Proof of Theorem bdinex2

Step	Hyp	Ref	Expression
1		incom 3329	. 2 |- ( B i^i A ) = ( A i^i B )
2		bdinex2.bd	. . 3 |- BOUNDED B
3		bdinex2.1	. . 3 |- A e. _V
4	2, 3	bdinex1 14690	. 2 |- ( A i^i B ) e. _V
5	1, 4	eqeltri 2250	1 |- ( B i^i A ) e. _V

Syntax hints:   e. wcel 2148   _Vcvv 2739   i^i cin 3130  BOUNDED wbdc 14631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsep 14675

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-bdc 14632

This theorem is referenced by:  bdssex  14693