Theorem ax13b 2031

Description: An equivalence between two ways of expressing ax-13 2377. See the comment for ax-13 2377. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)

Assertion

Ref	Expression
ax13b	⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑))))

Proof of Theorem ax13b

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ ((𝑦 = 𝑧 → 𝜑) → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑)))
2		equeuclr 2022	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
3	2	con3rr3 155	. . . . 5 ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ¬ 𝑥 = 𝑧))
4	3	imim1d 82	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ((¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑)) → (𝑦 = 𝑧 → (𝑦 = 𝑧 → 𝜑))))
5		pm2.43 56	. . . 4 ⊢ ((𝑦 = 𝑧 → (𝑦 = 𝑧 → 𝜑)) → (𝑦 = 𝑧 → 𝜑))
6	4, 5	syl6 35	. . 3 ⊢ (¬ 𝑥 = 𝑦 → ((¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑)) → (𝑦 = 𝑧 → 𝜑)))
7	1, 6	impbid2 226	. 2 ⊢ (¬ 𝑥 = 𝑦 → ((𝑦 = 𝑧 → 𝜑) ↔ (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑))))
8	7	pm5.74i 271	1 ⊢ ((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  ax13  2380  ax13ALT  2430  ax13fromc9  38907