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Metamath Proof Explorer

Theorem pm13.196a

Description: Theorem *13.196 in WhiteheadRussell p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011)

Ref Expression
Assertion pm13.196a 
|- ( -. ph <-> A. y ( [ y / x ] ph -> y =/= x ) )

Proof

Step Hyp Ref Expression
1 sbelx 
 |-  ( -. ph <-> E. y ( y = x /\ [ y / x ] -. ph ) )
2 sbalex 
 |-  ( E. y ( y = x /\ [ y / x ] -. ph ) <-> A. y ( y = x -> [ y / x ] -. ph ) )
3 sbn 
 |-  ( [ y / x ] -. ph <-> -. [ y / x ] ph )
4 3 imbi2i 
 |-  ( ( y = x -> [ y / x ] -. ph ) <-> ( y = x -> -. [ y / x ] ph ) )
5 con2b 
 |-  ( ( y = x -> -. [ y / x ] ph ) <-> ( [ y / x ] ph -> -. y = x ) )
6 df-ne 
 |-  ( y =/= x <-> -. y = x )
7 6 bicomi 
 |-  ( -. y = x <-> y =/= x )
8 7 imbi2i 
 |-  ( ( [ y / x ] ph -> -. y = x ) <-> ( [ y / x ] ph -> y =/= x ) )
9 4 5 8 3bitri 
 |-  ( ( y = x -> [ y / x ] -. ph ) <-> ( [ y / x ] ph -> y =/= x ) )
10 9 albii 
 |-  ( A. y ( y = x -> [ y / x ] -. ph ) <-> A. y ( [ y / x ] ph -> y =/= x ) )
11 1 2 10 3bitri 
 |-  ( -. ph <-> A. y ( [ y / x ] ph -> y =/= x ) )