Theorem bj-axemptylem 10683

Description: Lemma for bj-axempty 10684 and bj-axempty2 10685. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3904 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axemptylem	|- E. x A. y ( y e. x -> F. )

Distinct variable group:   x, y

Proof of Theorem bj-axemptylem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdfal 10624	. . 3 |- BOUNDED F.
2	1	bdsep1 10676	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ F. ) )
3		bi1 116	. . . 4 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> ( y e. z /\ F. ) ) )
4		falimd 1299	. . . 4 |- ( ( y e. z /\ F. ) -> F. )
5	3, 4	syl6 33	. . 3 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> F. ) )
6	5	alimi 1384	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y ( y e. x -> F. ) )
7	2, 6	eximii 1533	1 |- E. x A. y ( y e. x -> F. )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   F. wfal 1289   E.wex 1421

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

This theorem is referenced by:  bj-axempty  10684  bj-axempty2  10685