Theorem copsex4g

Description: An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995)

	Ref	Expression
Hypothesis	copsex4g.1	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	copsex4g	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ∧ 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		copsex4g.1	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ( 𝜑 ↔ 𝜓 ) )
2		eqcom	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
3		vex	⊢ 𝑥 ∈ V
4		vex	⊢ 𝑦 ∈ V
5	3 4	opth	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
6	2 5	bitri	⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) )
7		eqcom	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ↔ ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ )
8		vex	⊢ 𝑧 ∈ V
9		vex	⊢ 𝑤 ∈ V
10	8 9	opth	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) )
11	7 10	bitri	⊢ ( ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ↔ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) )
12	6 11	anbi12i	⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ↔ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
13	12	anbi1i	⊢ ( ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ∧ 𝜑 ) ↔ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ∧ 𝜑 ) )
14	13	a1i	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ∧ 𝜑 ) ↔ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ∧ 𝜑 ) ) )
15	14	4exbidv	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ∧ 𝜑 ) ) )
16		id	⊢ ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) )
17	16 1	cgsex4g	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ∧ ( 𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ) ) ∧ 𝜑 ) ↔ 𝜓 ) )
18	15 17	bitrd	⊢ ( ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ∧ ( 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆 ) ) → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ) ∧ 𝜑 ) ↔ 𝜓 ) )