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Metamath Proof Explorer

Theorem 2euex

Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2euexv when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

Ref Expression
Assertion 2euex 
|- ( 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 nfmo 
 |-  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 )