Theorem subsym1 32426

Description: A symmetry with [ x / y ].

See negsym1 32416 for more information. (Contributed by Anthony Hart, 11-Sep-2011.)

Assertion

Ref	Expression
subsym1	|- ( [ x / y ] [ x / y ] F. -> [ x / y ] ph )

Proof of Theorem subsym1

Step	Hyp	Ref	Expression
1		fal 1490	. . . . . . . . . 10 |- -. F.
2	1	intnan 960	. . . . . . . . 9 |- -. ( y = x /\ F. )
3	2	nex 1731	. . . . . . . 8 |- -. E. y ( y = x /\ F. )
4	3	intnan 960	. . . . . . 7 |- -. ( ( y = x -> F. ) /\ E. y ( y = x /\ F. ) )
5		df-sb 1881	. . . . . . 7 |- ( [ x / y ] F. <-> ( ( y = x -> F. ) /\ E. y ( y = x /\ F. ) ) )
6	4, 5	mtbir 313	. . . . . 6 |- -. [ x / y ] F.
7	6	intnan 960	. . . . 5 |- -. ( y = x /\ [ x / y ] F. )
8	7	nex 1731	. . . 4 |- -. E. y ( y = x /\ [ x / y ] F. )
9	8	intnan 960	. . 3 |- -. ( ( y = x -> [ x / y ] F. ) /\ E. y ( y = x /\ [ x / y ] F. ) )
10		df-sb 1881	. . 3 |- ( [ x / y ] [ x / y ] F. <-> ( ( y = x -> [ x / y ] F. ) /\ E. y ( y = x /\ [ x / y ] F. ) ) )
11	9, 10	mtbir 313	. 2 |- -. [ x / y ] [ x / y ] F.
12	11	pm2.21i 116	1 |- ( [ x / y ] [ x / y ] F. -> [ x / y ] ph )

Syntax hints:   -> wi 4   /\ wa 384   F. wfal 1488   E.wex 1704   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-sb 1881

This theorem is referenced by: (None)