Theorem sbequiOLD

Description: Obsolete proof of sbequi as of 7-Jul-2023. An equality theorem for substitution. (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 15-Sep-2018) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	sbequiOLD	$$⊢x=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		equtr	$$⊢z=x\to \left(x=y\to z=y\right)$$
2		sbequ2	$$⊢z=x\to \left(\left[x/z\right]\phi \to \phi \right)$$
3		sbequ1	$$⊢z=y\to \left(\phi \to \left[y/z\right]\phi \right)$$
4	2 3	syl9	$$⊢z=x\to \left(z=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)\right)$$
5	1 4	syld	$$⊢z=x\to \left(x=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)\right)$$
6		ax13	$$⊢\neg z=x\to \left(x=y\to \forall zx=y\right)$$
7		sp	$$⊢\forall zz=x\to z=x$$
8	7	con3i	$$⊢\neg z=x\to \neg \forall zz=x$$
9		sb4OLD	$$⊢\neg \forall zz=x\to \left(\left[x/z\right]\phi \to \forall z\left(z=x\to \phi \right)\right)$$
10	8 9	syl	$$⊢\neg z=x\to \left(\left[x/z\right]\phi \to \forall z\left(z=x\to \phi \right)\right)$$
11		equeuclr	$$⊢x=y\to \left(z=y\to z=x\right)$$
12	11	imim1d	$$⊢x=y\to \left(\left(z=x\to \phi \right)\to \left(z=y\to \phi \right)\right)$$
13	12	al2imi	$$⊢\forall zx=y\to \left(\forall z\left(z=x\to \phi \right)\to \forall z\left(z=y\to \phi \right)\right)$$
14		sb2	$$⊢\forall z\left(z=y\to \phi \right)\to \left[y/z\right]\phi$$
15	13 14	syl6	$$⊢\forall zx=y\to \left(\forall z\left(z=x\to \phi \right)\to \left[y/z\right]\phi \right)$$
16	10 15	syl9	$$⊢\neg z=x\to \left(\forall zx=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)\right)$$
17	6 16	syld	$$⊢\neg z=x\to \left(x=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)\right)$$
18	5 17	pm2.61i	$$⊢x=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)$$