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