Theorem wl-equsb3 33337

Description: equsb3 2432 with a distinctor. (Contributed by Wolf Lammen, 27-Jun-2019.)

Assertion

Ref	Expression
wl-equsb3	|- ( -. A. y y = z -> ( [ x / y ] y = z <-> x = z ) )

Proof of Theorem wl-equsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ w -. A. y y = z
2		nfna1 2029	. . . 4 |- F/ y -. A. y y = z
3		nfeqf2 2297	. . . 4 |- ( -. A. y y = z -> F/ y w = z )
4		equequ1 1952	. . . . 5 |- ( y = w -> ( y = z <-> w = z ) )
5	4	a1i 11	. . . 4 |- ( -. A. y y = z -> ( y = w -> ( y = z <-> w = z ) ) )
6	2, 3, 5	sbied 2409	. . 3 |- ( -. A. y y = z -> ( [ w / y ] y = z <-> w = z ) )
7	1, 6	sbbid 2403	. 2 |- ( -. A. y y = z -> ( [ x / w ] [ w / y ] y = z <-> [ x / w ] w = z ) )
8		sbcom3 2411	. . 3 |- ( [ x / w ] [ w / y ] y = z <-> [ x / w ] [ x / y ] y = z )
9		nfv 1843	. . . 4 |- F/ w [ x / y ] y = z
10	9	sbf 2380	. . 3 |- ( [ x / w ] [ x / y ] y = z <-> [ x / y ] y = z )
11	8, 10	bitri 264	. 2 |- ( [ x / w ] [ w / y ] y = z <-> [ x / y ] y = z )
12		equsb3 2432	. 2 |- ( [ x / w ] w = z <-> x = z )
13	7, 11, 12	3bitr3g 302	1 |- ( -. A. y y = z -> ( [ x / y ] y = z <-> x = z ) )

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: (None)