Theorem sbn 2391

Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

Assertion

Ref	Expression
sbn	|- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Proof of Theorem sbn

Step	Hyp	Ref	Expression
1		df-sb 1881	. . 3 |- ( [ y / x ] -. ph <-> ( ( x = y -> -. ph ) /\ E. x ( x = y /\ -. ph ) ) )
2		exanali 1786	. . . 4 |- ( E. x ( x = y /\ -. ph ) <-> -. A. x ( x = y -> ph ) )
3	2	anbi2i 730	. . 3 |- ( ( ( x = y -> -. ph ) /\ E. x ( x = y /\ -. ph ) ) <-> ( ( x = y -> -. ph ) /\ -. A. x ( x = y -> ph ) ) )
4		annim 441	. . 3 |- ( ( ( x = y -> -. ph ) /\ -. A. x ( x = y -> ph ) ) <-> -. ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) )
5	1, 3, 4	3bitri 286	. 2 |- ( [ y / x ] -. ph <-> -. ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) )
6		dfsb3 2374	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) )
7	5, 6	xchbinxr 325	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   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  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbi2  2393  sbor  2398  sban  2399  sbex  2463  sbcng  3476  difab  3896  bj-ab0  32902  wl-sb8et  33334  pm13.196a  38615