Theorem wl-sbcom3 33372

Description: Substituting y for x and then z for y is equivalent to substituting z for both x and y. Copy of ~? sbcom3OLD with a shortened proof.

Keep this theorem for a while here because an external reference to it exists.

(Contributed by Giovanni Mascellani, 8-Apr-2018.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
wl-sbcom3	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ z / y ] ph )

Proof of Theorem wl-sbcom3

Step	Hyp	Ref	Expression
1		nfa1 2028	. . . 4 |- F/ y A. y y = z
2		sbequ 2376	. . . . 5 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sps 2055	. . . 4 |- ( A. y y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
4	1, 3	sbbid 2403	. . 3 |- ( A. y y = z -> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
5	2	pm5.74i 260	. . . . . 6 |- ( ( y = z -> [ y / x ] ph ) <-> ( y = z -> [ z / x ] ph ) )
6	5	albii 1747	. . . . 5 |- ( A. y ( y = z -> [ y / x ] ph ) <-> A. y ( y = z -> [ z / x ] ph ) )
7	6	a1i 11	. . . 4 |- ( -. A. y y = z -> ( A. y ( y = z -> [ y / x ] ph ) <-> A. y ( y = z -> [ z / x ] ph ) ) )
8		sb4b 2358	. . . 4 |- ( -. A. y y = z -> ( [ z / y ] [ y / x ] ph <-> A. y ( y = z -> [ y / x ] ph ) ) )
9		sb4b 2358	. . . 4 |- ( -. A. y y = z -> ( [ z / y ] [ z / x ] ph <-> A. y ( y = z -> [ z / x ] ph ) ) )
10	7, 8, 9	3bitr4d 300	. . 3 |- ( -. A. y y = z -> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
11	4, 10	pm2.61i 176	. 2 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )
12		sbcom 2418	. 2 |- ( [ z / y ] [ z / x ] ph <-> [ z / x ] [ z / y ] ph )
13	11, 12	bitri 264	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ z / y ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   [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: (None)