Theorem wl-equsb4

Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable condition. (Contributed by Wolf Lammen, 26-Jun-2019)

	Ref	Expression
Assertion	wl-equsb4	$$⊢\neg \forall xx=z\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$

Proof

Step	Hyp	Ref	Expression
1		nfeqf	$$⊢\left(\neg \forall xx=y\wedge \neg \forall xx=z\right)\to Ⅎxy=z$$
2	1	ex	$$⊢\neg \forall xx=y\to \left(\neg \forall xx=z\to Ⅎxy=z\right)$$
3		sbft	$$⊢Ⅎxy=z\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$
4	2 3	syl6com	$$⊢\neg \forall xx=z\to \left(\neg \forall xx=y\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)\right)$$
5		sbequ12r	$$⊢y=x\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$
6	5	equcoms	$$⊢x=y\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$
7	6	sps	$$⊢\forall xx=y\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$
8	4 7	pm2.61d2	$$⊢\neg \forall xx=z\to \left(\left[y/x\right]y=z\leftrightarrow y=z\right)$$