Theorem latasymd

Description: Deduce equality from lattice ordering. ( eqssd analog.) (Contributed by NM, 18-Nov-2011)

	Ref	Expression
Hypotheses	latasymd.b	$$⊢B={\mathrm{Base}}_K$$
	latasymd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	latasymd.3	$$⊢\phi \to K\in \mathrm{Lat}$$
	latasymd.4	$$⊢\phi \to X\in B$$
	latasymd.5	$$⊢\phi \to Y\in B$$
	latasymd.6	$$⊢\phi \to X\underset{\dot{}}{\le }Y$$
	latasymd.7	$$⊢\phi \to Y\underset{\dot{}}{\le }X$$
Assertion	latasymd	$$⊢\phi \to X=Y$$

Proof

Step	Hyp	Ref	Expression
1		latasymd.b	$$⊢B={\mathrm{Base}}_K$$
2		latasymd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latasymd.3	$$⊢\phi \to K\in \mathrm{Lat}$$
4		latasymd.4	$$⊢\phi \to X\in B$$
5		latasymd.5	$$⊢\phi \to Y\in B$$
6		latasymd.6	$$⊢\phi \to X\underset{\dot{}}{\le }Y$$
7		latasymd.7	$$⊢\phi \to Y\underset{\dot{}}{\le }X$$
8	1 2	latasymb	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$
9	3 4 5 8	syl3anc	$$⊢\phi \to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$
10	6 7 9	mpbi2and	$$⊢\phi \to X=Y$$