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)

	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		nfv	⊢ Ⅎ 𝑧 𝜑
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	2 3 1	cbvopab1	⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } = { ⟨ 𝑧 , 𝑦 ⟩ ∣ 𝜓 }