Theorem isdomn5 40099

Description: The right conjunct in the right hand side of the equivalence of isdomn 20478 is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (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		bi2.04 388	. . . 4 ⊢ ((¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 )))
2		df-ne 2943	. . . . 5 ⊢ (𝑎 ≠ 0 ↔ ¬ 𝑎 = 0 )
3	2	imbi1i 349	. . . 4 ⊢ ((𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
4		df-or 844	. . . . 5 ⊢ ((𝑎 = 0 ∨ 𝑏 = 0 ) ↔ (¬ 𝑎 = 0 → 𝑏 = 0 ))
5	4	imbi2i 335	. . . 4 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 )))
6	1, 3, 5	3bitr4ri 303	. . 3 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
7	6	2ralbii 3091	. 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 3100	. . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
9	8	ralbii 3090	. 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 4726	. 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 296	1 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∨ wo 843   = wceq 1539   ≠ wne 2942  ∀wral 3063   ∖ cdif 3880  {csn 4558  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-nel 3049  df-ral 3068  df-v 3424  df-dif 3886  df-sn 4559

This theorem is referenced by:  isdomn4  40100