Theorem wl-spae 33306

Description: Prove an instance of sp 2053 from ax-13 2246 and Tarski's FOL only, without distinct variable conditions. The antecedent A. x x = y holds in a multi-object universe only if y is substituted for x, or vice versa, i.e. both variables are effectively the same. The converse -. A. x x = y indicates that both variables are distinct, and it so provides a simple translation of a distinct variable condition to a logical term. In case studies A. x x = y and -. A. x x = y can help eliminating distinct variable conditions.

The antecedent A. x x = y is expressed in the theorem's name by the abbreviation ae standing for 'all equal'.

Note that we cannot provide a logical predicate telling us directly whether a logical expression contains a particular variable, as such a construct would usually contradict ax-12 2047.

Note that this theorem is also provable from ax-12 2047 alone, so you can pick the axiom it is based on.

Compare this result to 19.3v 1897 and spaev 1978 having distinct variable conditions, but a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 5-Apr-2021.)

Assertion

Ref	Expression
wl-spae	|- ( A. x x = y -> x = y )

Proof of Theorem wl-spae

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aeveq 1982	. . . . 5 |- ( A. x x = z -> x = y )
2	1	adantl 482	. . . 4 |- ( ( y = z /\ A. x x = z ) -> x = y )
3	2	a1d 25	. . 3 |- ( ( y = z /\ A. x x = z ) -> ( A. x x = y -> x = y ) )
4		ax13v 2247	. . . . . . 7 |- ( -. x = y -> ( y = z -> A. x y = z ) )
5		equtrr 1949	. . . . . . . . 9 |- ( y = z -> ( x = y -> x = z ) )
6	5	al2imi 1743	. . . . . . . 8 |- ( A. x y = z -> ( A. x x = y -> A. x x = z ) )
7	6	con3d 148	. . . . . . 7 |- ( A. x y = z -> ( -. A. x x = z -> -. A. x x = y ) )
8	4, 7	syl6 35	. . . . . 6 |- ( -. x = y -> ( y = z -> ( -. A. x x = z -> -. A. x x = y ) ) )
9	8	com13 88	. . . . 5 |- ( -. A. x x = z -> ( y = z -> ( -. x = y -> -. A. x x = y ) ) )
10	9	impcom 446	. . . 4 |- ( ( y = z /\ -. A. x x = z ) -> ( -. x = y -> -. A. x x = y ) )
11	10	con4d 114	. . 3 |- ( ( y = z /\ -. A. x x = z ) -> ( A. x x = y -> x = y ) )
12	3, 11	pm2.61dan 832	. 2 |- ( y = z -> ( A. x x = y -> x = y ) )
13		ax6evr 1942	. 2 |- E. z y = z
14	12, 13	exlimiiv 1859	1 |- ( A. x x = y -> x = y )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481

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-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)