Theorem wl-spae

Description: Prove an instance of sp from ax-13 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 .

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

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

	Ref	Expression
Assertion	wl-spae	$$⊢\forall xx=y\to x=y$$

Proof

Step	Hyp	Ref	Expression
1		aeveq	$$⊢\forall xx=z\to x=y$$
2	1	adantl	$$⊢\left(y=z\wedge \forall xx=z\right)\to x=y$$
3	2	a1d	$$⊢\left(y=z\wedge \forall xx=z\right)\to \left(\forall xx=y\to x=y\right)$$
4		ax13v	$$⊢\neg x=y\to \left(y=z\to \forall xy=z\right)$$
5		equtrr	$$⊢y=z\to \left(x=y\to x=z\right)$$
6	5	al2imi	$$⊢\forall xy=z\to \left(\forall xx=y\to \forall xx=z\right)$$
7	6	con3d	$$⊢\forall xy=z\to \left(\neg \forall xx=z\to \neg \forall xx=y\right)$$
8	4 7	syl6	$$⊢\neg x=y\to \left(y=z\to \left(\neg \forall xx=z\to \neg \forall xx=y\right)\right)$$
9	8	com13	$$⊢\neg \forall xx=z\to \left(y=z\to \left(\neg x=y\to \neg \forall xx=y\right)\right)$$
10	9	impcom	$$⊢\left(y=z\wedge \neg \forall xx=z\right)\to \left(\neg x=y\to \neg \forall xx=y\right)$$
11	10	con4d	$$⊢\left(y=z\wedge \neg \forall xx=z\right)\to \left(\forall xx=y\to x=y\right)$$
12	3 11	pm2.61dan	$$⊢y=z\to \left(\forall xx=y\to x=y\right)$$
13		ax6evr	$$⊢\exists zy=z$$
14	12 13	exlimiiv	$$⊢\forall xx=y\to x=y$$