Theorem ax13b

Description: An equivalence between two ways of expressing ax-13 . See the comment for ax-13 . (Contributed by NM, 2-May-2017) (Proof shortened by Wolf Lammen, 26-Feb-2018) (Revised by BJ, 15-Sep-2020)

	Ref	Expression
Assertion	ax13b	$$⊢\left(\neg x=y\to \left(y=z\to \phi \right)\right)\leftrightarrow \left(\neg x=y\to \left(\neg x=z\to \left(y=z\to \phi \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\left(y=z\to \phi \right)\to \left(\neg x=z\to \left(y=z\to \phi \right)\right)$$
2		equeuclr	$$⊢y=z\to \left(x=z\to x=y\right)$$
3	2	con3rr3	$$⊢\neg x=y\to \left(y=z\to \neg x=z\right)$$
4	3	imim1d	$$⊢\neg x=y\to \left(\left(\neg x=z\to \left(y=z\to \phi \right)\right)\to \left(y=z\to \left(y=z\to \phi \right)\right)\right)$$
5		pm2.43	$$⊢\left(y=z\to \left(y=z\to \phi \right)\right)\to \left(y=z\to \phi \right)$$
6	4 5	syl6	$$⊢\neg x=y\to \left(\left(\neg x=z\to \left(y=z\to \phi \right)\right)\to \left(y=z\to \phi \right)\right)$$
7	1 6	impbid2	$$⊢\neg x=y\to \left(\left(y=z\to \phi \right)\leftrightarrow \left(\neg x=z\to \left(y=z\to \phi \right)\right)\right)$$
8	7	pm5.74i	$$⊢\left(\neg x=y\to \left(y=z\to \phi \right)\right)\leftrightarrow \left(\neg x=y\to \left(\neg x=z\to \left(y=z\to \phi \right)\right)\right)$$