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Metamath Proof Explorer

Theorem pm13.14

Description: Theorem *13.14 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011)

Ref Expression
Assertion pm13.14 
|- ( ( [. A / x ]. ph /\ -. ph ) -> x =/= A )

Proof

Step Hyp Ref Expression
1 sbceq1a 
 |-  ( x = A -> ( ph <-> [. A / x ]. ph ) )
2 1 biimprcd 
 |-  ( [. A / x ]. ph -> ( x = A -> ph ) )
3 2 necon3bd 
 |-  ( [. A / x ]. ph -> ( -. ph -> x =/= A ) )
4 3 imp 
 |-  ( ( [. A / x ]. ph /\ -. ph ) -> x =/= A )