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	|- ( ( -. x = y -> ( y = z -> ph ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	|- ( ( y = z -> ph ) -> ( -. x = z -> ( y = z -> ph ) ) )
2		equeuclr	|- ( y = z -> ( x = z -> x = y ) )
3	2	con3rr3	|- ( -. x = y -> ( y = z -> -. x = z ) )
4	3	imim1d	|- ( -. x = y -> ( ( -. x = z -> ( y = z -> ph ) ) -> ( y = z -> ( y = z -> ph ) ) ) )
5		pm2.43	|- ( ( y = z -> ( y = z -> ph ) ) -> ( y = z -> ph ) )
6	4 5	syl6	|- ( -. x = y -> ( ( -. x = z -> ( y = z -> ph ) ) -> ( y = z -> ph ) ) )
7	1 6	impbid2	|- ( -. x = y -> ( ( y = z -> ph ) <-> ( -. x = z -> ( y = z -> ph ) ) ) )
8	7	pm5.74i	|- ( ( -. x = y -> ( y = z -> ph ) ) <-> ( -. x = y -> ( -. x = z -> ( y = z -> ph ) ) ) )