Theorem pm13.196a 30856

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.)

Assertion

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

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem pm13.196a

Step	Hyp	Ref	Expression
1		sbelx 2185	. 2 |- ( -. ph <-> E. y ( y = x /\ [ y / x ] -. ph ) )
2		sb56 2149	. 2 |- ( E. y ( y = x /\ [ y / x ] -. ph ) <-> A. y ( y = x -> [ y / x ] -. ph ) )
3		sbn 2100	. . . . 5 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
4	3	imbi2i 312	. . . 4 |- ( ( y = x -> [ y / x ] -. ph ) <-> ( y = x -> -. [ y / x ] ph ) )
5		con2b 334	. . . 4 |- ( ( y = x -> -. [ y / x ] ph ) <-> ( [ y / x ] ph -> -. y = x ) )
6		df-ne 2659	. . . . . 6 |- ( y =/= x <-> -. y = x )
7	6	bicomi 202	. . . . 5 |- ( -. y = x <-> y =/= x )
8	7	imbi2i 312	. . . 4 |- ( ( [ y / x ] ph -> -. y = x ) <-> ( [ y / x ] ph -> y =/= x ) )
9	4, 5, 8	3bitri 271	. . 3 |- ( ( y = x -> [ y / x ] -. ph ) <-> ( [ y / x ] ph -> y =/= x ) )
10	9	albii 1615	. 2 |- ( A. y ( y = x -> [ y / x ] -. ph ) <-> A. y ( [ y / x ] ph -> y =/= x ) )
11	1, 2, 10	3bitri 271	1 |- ( -. ph <-> A. y ( [ y / x ] ph -> y =/= x ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1372   E.wex 1591   [wsb 1706   =/= wne 2657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-12 1798  ax-13 1963

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1592  df-nf 1595  df-sb 1707  df-ne 2659

This theorem is referenced by: (None)