Theorem equviniOLD

Description: Obsolete version of equvini as of 16-Sep-2023. (Contributed by NM, 10-Jan-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 15-Sep-2018) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	equviniOLD	$$⊢x=y\to \exists z\left(x=z\wedge z=y\right)$$

Proof

Step	Hyp	Ref	Expression
1		equtr	$$⊢z=x\to \left(x=y\to z=y\right)$$
2		equeuclr	$$⊢z=y\to \left(x=y\to x=z\right)$$
3	2	anc2ri	$$⊢z=y\to \left(x=y\to \left(x=z\wedge z=y\right)\right)$$
4	1 3	syli	$$⊢z=x\to \left(x=y\to \left(x=z\wedge z=y\right)\right)$$
5		19.8a	$$⊢\left(x=z\wedge z=y\right)\to \exists z\left(x=z\wedge z=y\right)$$
6	4 5	syl6	$$⊢z=x\to \left(x=y\to \exists z\left(x=z\wedge z=y\right)\right)$$
7		ax13	$$⊢\neg z=x\to \left(x=y\to \forall zx=y\right)$$
8		ax6e	$$⊢\exists zz=y$$
9	8 3	eximii	$$⊢\exists z\left(x=y\to \left(x=z\wedge z=y\right)\right)$$
10	9	19.35i	$$⊢\forall zx=y\to \exists z\left(x=z\wedge z=y\right)$$
11	7 10	syl6	$$⊢\neg z=x\to \left(x=y\to \exists z\left(x=z\wedge z=y\right)\right)$$
12	6 11	pm2.61i	$$⊢x=y\to \exists z\left(x=z\wedge z=y\right)$$