Theorem cbvopab2v

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999)

	Ref	Expression
Hypothesis	cbvopab2v.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvopab2v	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvopab2v.1	⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2		opeq2	⊢ ( 𝑦 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑧 ⟩ )
3	2	eqeq2d	⊢ ( 𝑦 = 𝑧 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ) )
4	3 1	anbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) ) )
5	4	cbvexvw	⊢ ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
6	5	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) )
7	6	abbii	⊢ { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
8		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
9		df-opab	⊢ { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑧 ( 𝑤 = ⟨ 𝑥 , 𝑧 ⟩ ∧ 𝜓 ) }
10	7 8 9	3eqtr4i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑥 , 𝑧 ⟩ ∣ 𝜓 }