Theorem bj-axemptylem 13774

Description: Lemma for bj-axempty 13775 and bj-axempty2 13776. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4108 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 13715	. . 3 |- BOUNDED F.
2	1	bdsep1 13767	. 2 |- E. x A. y ( y e. x <-> ( y e. z /\ F. ) )
3		biimp 117	. . . 4 |- ( ( y e. x <-> ( y e. z /\ F. ) ) -> ( y e. x -> ( y e. z /\ F. ) ) )
4		falimd 1358	. . . 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 1443	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y ( y e. x -> F. ) )
7	2, 6	eximii 1590	1 |- E. x A. y ( y e. x -> F. )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   F. wfal 1348   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-bd0 13695  ax-bdim 13696  ax-bdn 13699  ax-bdeq 13702  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349

This theorem is referenced by:  bj-axempty  13775  bj-axempty2  13776