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Metamath Proof Explorer

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 }