Theorem wl-cbvalnae

Description: A more general version of cbval when nonfree properties depend on a distinctor. Such expressions arise in proofs aiming at the elimination of distinct variable constraints, specifically in application of dvelimf , nfsb2 or dveeq1 . (Contributed by Wolf Lammen, 4-Jun-2019)

	Ref	Expression
Hypotheses	wl-cbvalnae.1	$$⊢\neg \forall xx=y\to Ⅎy\phi$$
	wl-cbvalnae.2	$$⊢\neg \forall xx=y\to Ⅎx\psi$$
	wl-cbvalnae.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	wl-cbvalnae	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		wl-cbvalnae.1	$$⊢\neg \forall xx=y\to Ⅎy\phi$$
2		wl-cbvalnae.2	$$⊢\neg \forall xx=y\to Ⅎx\psi$$
3		wl-cbvalnae.3	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
4		nftru	$$⊢Ⅎx\top$$
5		nftru	$$⊢Ⅎy\top$$
6	1	a1i	$$⊢\top \to \left(\neg \forall xx=y\to Ⅎy\phi \right)$$
7	2	a1i	$$⊢\top \to \left(\neg \forall xx=y\to Ⅎx\psi \right)$$
8	3	a1i	$$⊢\top \to \left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
9	4 5 6 7 8	wl-cbvalnaed	$$⊢\top \to \left(\forall x\phi \leftrightarrow \forall y\psi \right)$$
10	9	mptru	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$