Theorem sbequi 2375

Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)

Assertion

Ref	Expression
sbequi	|- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		equtr 1948	. . 3 |- ( z = x -> ( x = y -> z = y ) )
2		sbequ2 1882	. . . 4 |- ( z = x -> ( [ x / z ] ph -> ph ) )
3		sbequ1 2110	. . . 4 |- ( z = y -> ( ph -> [ y / z ] ph ) )
4	2, 3	syl9 77	. . 3 |- ( z = x -> ( z = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
5	1, 4	syld 47	. 2 |- ( z = x -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
6		ax13 2249	. . 3 |- ( -. z = x -> ( x = y -> A. z x = y ) )
7		sp 2053	. . . . . 6 |- ( A. z z = x -> z = x )
8	7	con3i 150	. . . . 5 |- ( -. z = x -> -. A. z z = x )
9		sb4 2356	. . . . 5 |- ( -. A. z z = x -> ( [ x / z ] ph -> A. z ( z = x -> ph ) ) )
10	8, 9	syl 17	. . . 4 |- ( -. z = x -> ( [ x / z ] ph -> A. z ( z = x -> ph ) ) )
11		equeuclr 1950	. . . . . . 7 |- ( x = y -> ( z = y -> z = x ) )
12	11	imim1d 82	. . . . . 6 |- ( x = y -> ( ( z = x -> ph ) -> ( z = y -> ph ) ) )
13	12	al2imi 1743	. . . . 5 |- ( A. z x = y -> ( A. z ( z = x -> ph ) -> A. z ( z = y -> ph ) ) )
14		sb2 2352	. . . . 5 |- ( A. z ( z = y -> ph ) -> [ y / z ] ph )
15	13, 14	syl6 35	. . . 4 |- ( A. z x = y -> ( A. z ( z = x -> ph ) -> [ y / z ] ph ) )
16	10, 15	syl9 77	. . 3 |- ( -. z = x -> ( A. z x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
17	6, 16	syld 47	. 2 |- ( -. z = x -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
18	5, 17	pm2.61i 176	1 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   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:  sbequ  2376