Theorem bdinex2 16670

Description: Bounded version of inex2 4245. (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 3411	. 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 16669	. 2 |- ( A i^i B ) e. _V
5	1, 4	eqeltri 2305	1 |- ( B i^i A ) e. _V

Syntax hints:   e. wcel 2203   _Vcvv 2813   i^i cin 3210  BOUNDED wbdc 16610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214  ax-bdsep 16654

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-bdc 16611

This theorem is referenced by:  bdssex  16672