Theorem bj-axempty 15539

Description: Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4158. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4159 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axempty	|- E. x A. y e. x F.

Distinct variable group:   x, y

Proof of Theorem bj-axempty

Step	Hyp	Ref	Expression
1		bj-axemptylem 15538	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2480	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1619	. 2 |- ( E. x A. y e. x F. <-> E. x A. y ( y e. x -> F. ) )
4	1, 3	mpbir 146	1 |- E. x A. y e. x F.

Syntax hints:   -> wi 4   A.wal 1362   F. wfal 1369   E.wex 1506   A.wral 2475

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-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548  ax-bd0 15459  ax-bdim 15460  ax-bdn 15463  ax-bdeq 15466  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-ral 2480

This theorem is referenced by: (None)