Theorem prneimg2 4809

Description: Two pairs are not equal if their counterparts are not equal. (Contributed by AV, 5-Sep-2025.)

Assertion

Ref	Expression
prneimg2	⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶))))

Proof of Theorem prneimg2

Step	Hyp	Ref	Expression
1		preq12bg 4807	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2	1	necon3abid 2991	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
3		ioran 999	. . 3 ⊢ (¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
4		ianor 997	. . . . 5 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5		df-ne 2956	. . . . . 6 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
6		df-ne 2956	. . . . . 6 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷)
7	5, 6	orbi12i 928	. . . . 5 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
8	4, 7	bitr4i 281	. . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
9		ianor 997	. . . . 5 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶))
10		df-ne 2956	. . . . . 6 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
11		df-ne 2956	. . . . . 6 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
12	10, 11	orbi12i 928	. . . . 5 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶))
13	9, 12	bitr4i 281	. . . 4 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶))
14	8, 13	anbi12i 640	. . 3 ⊢ ((¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶)))
15	3, 14	bitri 278	. 2 ⊢ (¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶)))
16	2, 15	bitrdi 290	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-un 4150  df-sn 4585  df-pr 4587

This theorem is used by:  gpg5nbgrvtx03starlem1  49273  gpg5nbgrvtx03starlem2  49274  gpg5nbgrvtx03starlem3  49275  gpg5nbgrvtx13starlem1  49276  gpg5nbgrvtx13starlem2  49277  gpg5nbgrvtx13starlem3  49278  gpg5edgnedg  49335