This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb9

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) Allow a shortening of sb9i . (Revised by Wolf Lammen, 15-Jun-2019) (New usage is discouraged.)

Ref Expression
Assertion sb9 
|- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Proof

Step Hyp Ref Expression
1 sbequ12a 
 |-  ( y = x -> ( [ x / y ] ph <-> [ y / x ] ph ) )
2 1 equcoms 
 |-  ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
3 2 sps 
 |-  ( A. x x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
4 3 dral1 
 |-  ( A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
5 nfnae 
 |-  F/ x -. A. x x = y
6 nfnae 
 |-  F/ y -. A. x x = y
7 nfsb2 
 |-  ( -. A. y y = x -> F/ y [ x / y ] ph )
8 7 naecoms 
 |-  ( -. A. x x = y -> F/ y [ x / y ] ph )
9 nfsb2 
 |-  ( -. A. x x = y -> F/ x [ y / x ] ph )
10 2 a1i 
 |-  ( -. A. x x = y -> ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) ) )
11 5 6 8 9 10 cbv2 
 |-  ( -. A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
12 4 11 pm2.61i 
 |-  ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )