Theorem sbcom3 2411

Description: Substituting y for x and then z for y is equivalent to substituting z for both x and y. (Contributed by Giovanni Mascellani, 8-Apr-2018.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 16-Sep-2018.) (Proof shortened by Wolf Lammen, 16-Sep-2018.)

Assertion

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

Proof of Theorem sbcom3

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ y A. y y = z
2		drsb2 2378	. . 3 |- ( A. y y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	1, 2	sbbid 2403	. 2 |- ( A. y y = z -> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
4		sb4b 2358	. . . 4 |- ( -. A. y y = z -> ( [ z / y ] [ y / x ] ph <-> A. y ( y = z -> [ y / x ] ph ) ) )
5		sbequ 2376	. . . . . 6 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
6	5	pm5.74i 260	. . . . 5 |- ( ( y = z -> [ y / x ] ph ) <-> ( y = z -> [ z / x ] ph ) )
7	6	albii 1747	. . . 4 |- ( A. y ( y = z -> [ y / x ] ph ) <-> A. y ( y = z -> [ z / x ] ph ) )
8	4, 7	syl6bb 276	. . 3 |- ( -. A. y y = z -> ( [ z / y ] [ y / x ] ph <-> A. y ( y = z -> [ z / x ] ph ) ) )
9		sb4b 2358	. . 3 |- ( -. A. y y = z -> ( [ z / y ] [ z / x ] ph <-> A. y ( y = z -> [ z / x ] ph ) ) )
10	8, 9	bitr4d 271	. 2 |- ( -. A. y y = z -> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
11	3, 10	pm2.61i 176	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] 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-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:  sbco  2412  sbidm  2414  sbcom  2418  equsb3  2432  wl-equsb3  33337