Theorem bj-axemptylem 14915

Description: Lemma for bj-axempty 14916 and bj-axempty2 14917. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 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 14856	. . 3 |- BOUNDED F.
2	1	bdsep1 14908	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ F. ) )
3		biimp 118	. . . 4 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> ( y e. z /\ F. ) ) )
4		falimd 1378	. . . 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 1465	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y ( y e. x -> F. ) )
7	2, 6	eximii 1612	1 |- E. x A. y ( y e. x -> F. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1361   F. wfal 1368   E.wex 1502

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544  ax-bd0 14836  ax-bdim 14837  ax-bdn 14840  ax-bdeq 14843  ax-bdsep 14907

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369

This theorem is referenced by:  bj-axempty  14916  bj-axempty2  14917