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Metamath Proof Explorer

Theorem pm14.12

Description: Theorem *14.12 in WhiteheadRussell p. 184. (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion pm14.12 
|- ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )

Proof

Step Hyp Ref Expression
1 eumo 
 |-  ( E! x ph -> E* x ph )
2 nfv 
 |-  F/ y ph
3 2 mo3 
 |-  ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
4 sbsbc 
 |-  ( [ y / x ] ph <-> [. y / x ]. ph )
5 4 anbi2i 
 |-  ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ [. y / x ]. ph ) )
6 5 imbi1i 
 |-  ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
7 6 2albii 
 |-  ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
8 3 7 bitri 
 |-  ( E* x ph <-> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )
9 1 8 sylib 
 |-  ( E! x ph -> A. x A. y ( ( ph /\ [. y / x ]. ph ) -> x = y ) )