Theorem vopelopabsb

Description: The law of concretion in terms of substitutions. Version of opelopabsb with set variables. (Contributed by NM, 30-Sep-2002) (Proof shortened by Andrew Salmon, 25-Jul-2011) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024)

	Ref	Expression
Assertion	vopelopabsb	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ )
2		vex	⊢ 𝑥 ∈ V
3		vex	⊢ 𝑦 ∈ V
4	2 3	opth	⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑤 ⟩ ↔ ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) )
5	1 4	bitri	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ↔ ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) )
6	5	anbi1i	⊢ ( ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )
7	6	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )
8		elopab	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝑧 , 𝑤 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
9		2sb5	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ∧ 𝜑 ) )
10	7 8 9	3bitr4i	⊢ ( ⟨ 𝑧 , 𝑤 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 )