Theorem sbcom2v 2004

Description: Lemma for proving sbcom2 2006. It is the same as sbcom2 2006 but with additional distinct variable constraints on x and y, and on w and z. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2v	|- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Distinct variable groups:   x, w, z   x, y, z

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem sbcom2v

Step	Hyp	Ref	Expression
1		alcom 1492	. . 3 |- ( A. z A. x ( x = y -> ( z = w -> ph ) ) <-> A. x A. z ( x = y -> ( z = w -> ph ) ) )
2		bi2.04 248	. . . . . 6 |- ( ( x = y -> ( z = w -> ph ) ) <-> ( z = w -> ( x = y -> ph ) ) )
3	2	albii 1484	. . . . 5 |- ( A. x ( x = y -> ( z = w -> ph ) ) <-> A. x ( z = w -> ( x = y -> ph ) ) )
4		19.21v 1887	. . . . 5 |- ( A. x ( z = w -> ( x = y -> ph ) ) <-> ( z = w -> A. x ( x = y -> ph ) ) )
5	3, 4	bitri 184	. . . 4 |- ( A. x ( x = y -> ( z = w -> ph ) ) <-> ( z = w -> A. x ( x = y -> ph ) ) )
6	5	albii 1484	. . 3 |- ( A. z A. x ( x = y -> ( z = w -> ph ) ) <-> A. z ( z = w -> A. x ( x = y -> ph ) ) )
7		19.21v 1887	. . . 4 |- ( A. z ( x = y -> ( z = w -> ph ) ) <-> ( x = y -> A. z ( z = w -> ph ) ) )
8	7	albii 1484	. . 3 |- ( A. x A. z ( x = y -> ( z = w -> ph ) ) <-> A. x ( x = y -> A. z ( z = w -> ph ) ) )
9	1, 6, 8	3bitr3i 210	. 2 |- ( A. z ( z = w -> A. x ( x = y -> ph ) ) <-> A. x ( x = y -> A. z ( z = w -> ph ) ) )
10		sb6 1901	. . 3 |- ( [ w / z ] [ y / x ] ph <-> A. z ( z = w -> [ y / x ] ph ) )
11		sb6 1901	. . . . 5 |- ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
12	11	imbi2i 226	. . . 4 |- ( ( z = w -> [ y / x ] ph ) <-> ( z = w -> A. x ( x = y -> ph ) ) )
13	12	albii 1484	. . 3 |- ( A. z ( z = w -> [ y / x ] ph ) <-> A. z ( z = w -> A. x ( x = y -> ph ) ) )
14	10, 13	bitri 184	. 2 |- ( [ w / z ] [ y / x ] ph <-> A. z ( z = w -> A. x ( x = y -> ph ) ) )
15		sb6 1901	. . 3 |- ( [ y / x ] [ w / z ] ph <-> A. x ( x = y -> [ w / z ] ph ) )
16		sb6 1901	. . . . 5 |- ( [ w / z ] ph <-> A. z ( z = w -> ph ) )
17	16	imbi2i 226	. . . 4 |- ( ( x = y -> [ w / z ] ph ) <-> ( x = y -> A. z ( z = w -> ph ) ) )
18	17	albii 1484	. . 3 |- ( A. x ( x = y -> [ w / z ] ph ) <-> A. x ( x = y -> A. z ( z = w -> ph ) ) )
19	15, 18	bitri 184	. 2 |- ( [ y / x ] [ w / z ] ph <-> A. x ( x = y -> A. z ( z = w -> ph ) ) )
20	9, 14, 19	3bitr4i 212	1 |- ( [ w / z ] [ y / x ] ph <-> [ y / x ] [ w / z ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   [wsb 1776

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  sbcom2v2  2005