Theorem isdomn5 20820

Description: The equivalence between the right conjuncts in the right hand sides of isdomn 20815 and isdomn2 20821, in predicate calculus form. (Contributed by SN, 16-Sep-2024.)

Assertion

Ref	Expression
isdomn5	⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))

Distinct variable group:   0 ,𝑎,𝑏

Allowed substitution hints:   𝐵(𝑎, 𝑏)   · (𝑎, 𝑏)

Proof of Theorem isdomn5

Step	Hyp	Ref	Expression
1		bi2.04 391	. . . 4 ⊢ ((¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 )))
2		df-ne 2958	. . . . 5 ⊢ (𝑎 ≠ 0 ↔ ¬ 𝑎 = 0 )
3	2	imbi1i 352	. . . 4 ⊢ ((𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
4		df-or 861	. . . . 5 ⊢ ((𝑎 = 0 ∨ 𝑏 = 0 ) ↔ (¬ 𝑎 = 0 → 𝑏 = 0 ))
5	4	imbi2i 339	. . . 4 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 )))
6	1, 3, 5	3bitr4ri 307	. . 3 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
7	6	2ralbii 3139	. 2 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
8		r19.21v 3189	. . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
9	8	ralbii 3110	. 2 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ∀𝑎 ∈ 𝐵 (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
10		raldifsnb 4763	. 2 ⊢ (∀𝑎 ∈ 𝐵 (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))
11	7, 9, 10	3bitri 300	1 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1569   ≠ wne 2957  ∀wral 3078   ∖ cdif 3901  {csn 4588  (class class class)co 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-ral 3079  df-v 3456  df-dif 3907  df-sn 4589

This theorem is used by:  isdomn2  20821  isdomn4  20825