Theorem wl-sbal1

Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor -. A. x x = z . (Contributed by NM, 15-May-1993) Proof is based on wl-sbalnae now. See also sbal1 . (Revised by Wolf Lammen, 25-Jul-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	wl-sbal1	$$⊢\neg \forall xx=z\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		naev	$$⊢\neg \forall xx=z\to \neg \forall xx=y$$
2		wl-sbalnae	$$⊢\left(\neg \forall xx=y\wedge \neg \forall xx=z\right)\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi \right)$$
3	1 2	mpancom	$$⊢\neg \forall xx=z\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi \right)$$