Theorem cbvopab1v

Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003) (Proof shortened by Eric Schmidt, 4-Apr-2007) Reduce axiom usage. (Revised by GG, 17-Nov-2024)

	Ref	Expression
Hypothesis	cbvopab1v.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvopab1v	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvopab1v.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2		opeq1	⊢ ( 𝑥 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑧 , 𝑦 ⟩ )
3	2	eqeq2d	⊢ ( 𝑥 = 𝑧 → ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ↔ 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ) )
4	3 1	anbi12d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) ) )
5	4	exbidv	⊢ ( 𝑥 = 𝑧 → ( ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) ) )
6	5	cbvexvw	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) )
7	6	abbii	⊢ { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) } = { 𝑤 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) }
8		df-opab	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑤 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) }
9		df-opab	⊢ { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑧 ∃ 𝑦 ( 𝑤 = ⟨ 𝑧 , 𝑦 ⟩ ∧ 𝜓 ) }
10	7 8 9	3eqtr4i	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 }