Theorem prneimg2

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

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

Proof

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