Theorem sb9 2030

Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)

Assertion

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

Proof of Theorem sb9

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sb9v 2029	. . 3 |- ( A. y [ y / w ] [ w / x ] ph <-> A. w [ w / y ] [ w / x ] ph )
2		sbcom 2026	. . . 4 |- ( [ w / y ] [ w / x ] ph <-> [ w / x ] [ w / y ] ph )
3	2	albii 1516	. . 3 |- ( A. w [ w / y ] [ w / x ] ph <-> A. w [ w / x ] [ w / y ] ph )
4		sb9v 2029	. . 3 |- ( A. w [ w / x ] [ w / y ] ph <-> A. x [ x / w ] [ w / y ] ph )
5	1, 3, 4	3bitri 206	. 2 |- ( A. y [ y / w ] [ w / x ] ph <-> A. x [ x / w ] [ w / y ] ph )
6		ax-17 1572	. . . 4 |- ( ph -> A. w ph )
7	6	sbco2h 2015	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	albii 1516	. 2 |- ( A. y [ y / w ] [ w / x ] ph <-> A. y [ y / x ] ph )
9	6	sbco2h 2015	. . 3 |- ( [ x / w ] [ w / y ] ph <-> [ x / y ] ph )
10	9	albii 1516	. 2 |- ( A. x [ x / w ] [ w / y ] ph <-> A. x [ x / y ] ph )
11	5, 8, 10	3bitr3ri 211	1 |- ( A. x [ x / y ] ph <-> A. y [ y / x ] ph )

Syntax hints:   <-> wb 105   A.wal 1393   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sb9i  2031