Theorem equid1ALT

Description: Alternate proof of equid and equid1 from older axioms ax-c7 , ax-c10 and ax-c9 . (Contributed by NM, 10-Jan-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	equid1ALT	$$⊢x=x$$

Proof

Step	Hyp	Ref	Expression
1		ax-c9	$$⊢\neg \forall xx=x\to \left(\neg \forall xx=x\to \left(x=x\to \forall xx=x\right)\right)$$
2	1	pm2.43i	$$⊢\neg \forall xx=x\to \left(x=x\to \forall xx=x\right)$$
3	2	alimi	$$⊢\forall x\neg \forall xx=x\to \forall x\left(x=x\to \forall xx=x\right)$$
4		ax-c10	$$⊢\forall x\left(x=x\to \forall xx=x\right)\to x=x$$
5	3 4	syl	$$⊢\forall x\neg \forall xx=x\to x=x$$
6		ax-c7	$$⊢\neg \forall x\neg \forall xx=x\to x=x$$
7	5 6	pm2.61i	$$⊢x=x$$