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 → 𝜑 ) ) ↔ ( ¬ 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( ( 𝑦 = 𝑧 → 𝜑 ) → ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) )
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 → 𝜑 ) ) → ( 𝑦 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) ) )
5		pm2.43	⊢ ( ( 𝑦 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) → ( 𝑦 = 𝑧 → 𝜑 ) )
6	4 5	syl6	⊢ ( ¬ 𝑥 = 𝑦 → ( ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) → ( 𝑦 = 𝑧 → 𝜑 ) ) )
7	1 6	impbid2	⊢ ( ¬ 𝑥 = 𝑦 → ( ( 𝑦 = 𝑧 → 𝜑 ) ↔ ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) ) )
8	7	pm5.74i	⊢ ( ( ¬ 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → 𝜑 ) ) ↔ ( ¬ 𝑥 = 𝑦 → ( ¬ 𝑥 = 𝑧 → ( 𝑦 = 𝑧 → 𝜑 ) ) ) )