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Metamath Proof Explorer

Theorem pm13.192

Description: Theorem *13.192 in WhiteheadRussell p. 179. (Contributed by Andrew Salmon, 3-Jun-2011) (Revised by NM, 4-Jan-2017)

Ref Expression
Assertion pm13.192 
|- ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> [. A / y ]. ph )

Proof

Step Hyp Ref Expression
1 biimpr 
 |-  ( ( x = A <-> x = y ) -> ( x = y -> x = A ) )
2 1 alimi 
 |-  ( A. x ( x = A <-> x = y ) -> A. x ( x = y -> x = A ) )
3 eqeq1 
 |-  ( x = y -> ( x = A <-> y = A ) )
4 3 equsalvw 
 |-  ( A. x ( x = y -> x = A ) <-> y = A )
5 2 4 sylib 
 |-  ( A. x ( x = A <-> x = y ) -> y = A )
6 eqeq2 
 |-  ( A = y -> ( x = A <-> x = y ) )
7 6 eqcoms 
 |-  ( y = A -> ( x = A <-> x = y ) )
8 7 alrimiv 
 |-  ( y = A -> A. x ( x = A <-> x = y ) )
9 5 8 impbii 
 |-  ( A. x ( x = A <-> x = y ) <-> y = A )
10 9 anbi1i 
 |-  ( ( A. x ( x = A <-> x = y ) /\ ph ) <-> ( y = A /\ ph ) )
11 10 exbii 
 |-  ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> E. y ( y = A /\ ph ) )
12 sbc5 
 |-  ( [. A / y ]. ph <-> E. y ( y = A /\ ph ) )
13 11 12 bitr4i 
 |-  ( E. y ( A. x ( x = A <-> x = y ) /\ ph ) <-> [. A / y ]. ph )