This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2euexv

Description: Double quantification with existential uniqueness. Version of 2euex with x and y distinct, but not requiring ax-13 . (Contributed by NM, 3-Dec-2001) (Revised by Wolf Lammen, 2-Oct-2023)

Ref Expression
Assertion 2euexv 
|- ( E! x E. y ph -> E. y E! x ph )

Proof

Step Hyp Ref Expression
1 df-eu 
 |-  ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2 excom 
 |-  ( E. x E. y ph <-> E. y E. x ph )
3 nfe1 
 |-  F/ y E. y ph
4 3 nfmov 
 |-  F/ y E* x E. y ph
5 19.8a 
 |-  ( ph -> E. y ph )
6 5 moimi 
 |-  ( E* x E. y ph -> E* x ph )
7 moeu 
 |-  ( E* x ph <-> ( E. x ph -> E! x ph ) )
8 6 7 sylib 
 |-  ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9 4 8 eximd 
 |-  ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10 2 9 biimtrid 
 |-  ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11 10 impcom 
 |-  ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12 1 11 sylbi 
 |-  ( E! x E. y ph -> E. y E! x ph )