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 ) )