Theorem prneimg2 4880

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 4878	. . 3 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
2	1	necon3abid 2983	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶))))
3		ioran 984	. . 3 ⊢ (¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)))
4		ianor 982	. . . . 5 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
5		df-ne 2947	. . . . . 6 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶)
6		df-ne 2947	. . . . . 6 ⊢ (𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷)
7	5, 6	orbi12i 913	. . . . 5 ⊢ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ↔ (¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷))
8	4, 7	bitr4i 278	. . . 4 ⊢ (¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
9		ianor 982	. . . . 5 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶))
10		df-ne 2947	. . . . . 6 ⊢ (𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷)
11		df-ne 2947	. . . . . 6 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶)
12	10, 11	orbi12i 913	. . . . 5 ⊢ ((𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶) ↔ (¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶))
13	9, 12	bitr4i 278	. . . 4 ⊢ (¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶) ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶))
14	8, 13	anbi12i 627	. . 3 ⊢ ((¬ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ ¬ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶)))
15	3, 14	bitri 275	. 2 ⊢ (¬ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∨ (𝐴 = 𝐷 ∧ 𝐵 = 𝐶)) ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶)))
16	2, 15	bitrdi 287	1 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} ↔ ((𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷) ∧ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by:  gpg5nbgrvtx03starlem1  47911  gpg5nbgrvtx03starlem2  47912  gpg5nbgrvtx03starlem3  47913  gpg5nbgrvtx13starlem1  47914  gpg5nbgrvtx13starlem2  47915  gpg5nbgrvtx13starlem3  47916