Theorem bj-axempty 10684

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 non-empty universe. See axnul 3903. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 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 10683	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2353	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1536	. 2 |- ( E. x A. y e. x F. <-> E. x A. y ( y e. x -> F. ) )
4	1, 3	mpbir 144	1 |- E. x A. y e. x F.

Syntax hints:   -> wi 4   A.wal 1282   F. wfal 1289   E.wex 1421   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-bd0 10604  ax-bdim 10605  ax-bdn 10608  ax-bdeq 10611  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-ral 2353

This theorem is referenced by: (None)