Theorem equvini

Description: A variable introduction law for equality. Lemma 15 of Monk2 p. 109, however we do not require z to be distinct from x and y . Usage of this theorem is discouraged because it depends on ax-13 . See equvinv for a shorter proof requiring fewer axioms when z is required to be distinct from x and y . (Contributed by NM, 10-Jan-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 16-Sep-2023) (New usage is discouraged.)

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

Proof

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