Theorem sbequivvOLD

Description: Obsolete version of sbequi as of 7-Jul-2023. Version of sbequi with disjoint variable conditions, not requiring ax-13 . (Contributed by Wolf Lammen, 19-Jan-2023) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		equeuclr	$$⊢x=y\to \left(z=y\to z=x\right)$$
2	1	imim1d	$$⊢x=y\to \left(\left(z=x\to \phi \right)\to \left(z=y\to \phi \right)\right)$$
3	2	alimdv	$$⊢x=y\to \left(\forall z\left(z=x\to \phi \right)\to \forall z\left(z=y\to \phi \right)\right)$$
4		sb6	$$⊢\left[x/z\right]\phi \leftrightarrow \forall z\left(z=x\to \phi \right)$$
5		sb6	$$⊢\left[y/z\right]\phi \leftrightarrow \forall z\left(z=y\to \phi \right)$$
6	3 4 5	3imtr4g	$$⊢x=y\to \left(\left[x/z\right]\phi \to \left[y/z\right]\phi \right)$$