Theorem isdomn5 20872

Description: The equivalence between the right conjuncts in the right hand sides of isdomn 20867 and isdomn2 20873, 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		bi2.04 392	. . . 4 ⊢ ((¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 )))
2		df-ne 2956	. . . . 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 862	. . . . 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 3137	. 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 3187	. . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )))
9	8	ralbii 3108	. 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 4759	. 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 861   = wceq 1570   ≠ wne 2955  ∀wral 3076   ∖ cdif 3896  {csn 4584  (class class class)co 7413

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-ral 3077  df-v 3452  df-dif 3902  df-sn 4585

This theorem is used by:  isdomn2  20873  isdomn4  20877