Theorem bj-axemptylem 10012

Description: Lemma for bj-axempty 10013 and bj-axempty2 10014. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 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 9953	. . 3 |- BOUNDED F.
2	1	bdsep1 10005	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ F. ) )
3		bi1 111	. . . 4 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> ( y e. z /\ F. ) ) )
4		falimd 1258	. . . 4 |- ( ( y e. z /\ F. ) -> F. )
5	3, 4	syl6 29	. . 3 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> F. ) )
6	5	alimi 1344	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y ( y e. x -> F. ) )
7	2, 6	eximii 1493	1 |- E. x A. y ( y e. x -> F. )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   F. wfal 1248   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by:  bj-axempty  10013  bj-axempty2  10014