Theorem bj-axemptylem 13427

Description: Lemma for bj-axempty 13428 and bj-axempty2 13429. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4090 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 13368	. . 3 |- BOUNDED F.
2	1	bdsep1 13420	. 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 1350	. . . 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 1435	. 2 |- ( A. y ( y e. x <-> ( y e. z /\ F. ) ) -> A. y ( y e. x -> F. ) )
7	2, 6	eximii 1582	1 |- E. x A. y ( y e. x -> F. )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   F. wfal 1340   E.wex 1472

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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514  ax-bd0 13348  ax-bdim 13349  ax-bdn 13352  ax-bdeq 13355  ax-bdsep 13419

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341

This theorem is referenced by:  bj-axempty  13428  bj-axempty2  13429