Theorem wl-sbal2

Description: Move quantifier in and out of substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 2-Jan-2002) Proof is based on wl-sbalnae now. See also sbal2 . (Revised by Wolf Lammen, 25-Jul-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	wl-sbal2	$$⊢\neg \forall xx=y\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		naev	$$⊢\neg \forall xx=y\to \neg \forall xx=z$$
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	mpdan	$$⊢\neg \forall xx=y\to \left(\left[z/y\right]\forall x\phi \leftrightarrow \forall x\left[z/y\right]\phi \right)$$