Theorem sb9 2426

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2427. (Revised by Wolf Lammen, 15-Jun-2019.)

Assertion

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

Proof of Theorem sb9

Step	Hyp	Ref	Expression
1		sbequ12a 2113	. . . . 5 |- ( y = x -> ( [ x / y ] ph <-> [ y / x ] ph ) )
2	1	equcoms 1947	. . . 4 |- ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
3	2	sps 2055	. . 3 |- ( A. x x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) )
4	3	dral1 2325	. 2 |- ( A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
5		nfnae 2318	. . 3 |- F/ x -. A. x x = y
6		nfnae 2318	. . 3 |- F/ y -. A. x x = y
7		nfsb2 2360	. . . 4 |- ( -. A. y y = x -> F/ y [ x / y ] ph )
8	7	naecoms 2313	. . 3 |- ( -. A. x x = y -> F/ y [ x / y ] ph )
9		nfsb2 2360	. . 3 |- ( -. A. x x = y -> F/ x [ y / x ] ph )
10	2	a1i 11	. . 3 |- ( -. A. x x = y -> ( x = y -> ( [ x / y ] ph <-> [ y / x ] ph ) ) )
11	5, 6, 8, 9, 10	cbv2 2270	. 2 |- ( -. A. x x = y -> ( A. x [ x / y ] ph <-> A. y [ y / x ] ph ) )
12	4, 11	pm2.61i 176	1 |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708   [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-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sb9i  2427