Theorem cbvoprab3v

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypothesis	cbvoprab3v.1	⊢ ( 𝑧 = 𝑤 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvoprab3v	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑤 ⟩ ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvoprab3v.1	⊢ ( 𝑧 = 𝑤 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑤 𝜑
3		nfv	⊢ Ⅎ 𝑧 𝜓
4	2 3 1	cbvoprab3	⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑤 ⟩ ∣ 𝜓 }