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Metamath Proof Explorer

Theorem eu1

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of Margaris p. 110. (Contributed by NM, 20-Aug-1993) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 29-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 7-Feb-2023)

Ref Expression
Hypothesis eu1.nf 
|- F/ y ph
Assertion eu1 
|- ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )

Proof

Step Hyp Ref Expression
1 eu1.nf 
 |-  F/ y ph
2 nfs1v 
 |-  F/ x [ y / x ] ph
3 2 euf 
 |-  ( E! y [ y / x ] ph <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
4 1 sb8euv 
 |-  ( E! x ph <-> E! y [ y / x ] ph )
5 1 sb6rfv 
 |-  ( ph <-> A. y ( y = x -> [ y / x ] ph ) )
6 equcom 
 |-  ( x = y <-> y = x )
7 6 imbi2i 
 |-  ( ( [ y / x ] ph -> x = y ) <-> ( [ y / x ] ph -> y = x ) )
8 7 albii 
 |-  ( A. y ( [ y / x ] ph -> x = y ) <-> A. y ( [ y / x ] ph -> y = x ) )
9 5 8 anbi12ci 
 |-  ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
10 albiim 
 |-  ( A. y ( [ y / x ] ph <-> y = x ) <-> ( A. y ( [ y / x ] ph -> y = x ) /\ A. y ( y = x -> [ y / x ] ph ) ) )
11 9 10 bitr4i 
 |-  ( ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> A. y ( [ y / x ] ph <-> y = x ) )
12 11 exbii 
 |-  ( E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) <-> E. x A. y ( [ y / x ] ph <-> y = x ) )
13 3 4 12 3bitr4i 
 |-  ( E! x ph <-> E. x ( ph /\ A. y ( [ y / x ] ph -> x = y ) ) )