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Metamath Proof Explorer

Theorem oprabbid

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004) (Revised by Mario Carneiro, 24-Jun-2014)

Ref Expression
Hypotheses oprabbid.1 ⊢ Ⅎ 𝑥 𝜑
oprabbid.2 ⊢ Ⅎ 𝑦 𝜑
oprabbid.3 ⊢ Ⅎ 𝑧 𝜑
oprabbid.4 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion oprabbid ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } )

Proof

Step Hyp Ref Expression
1 oprabbid.1 ⊢ Ⅎ 𝑥 𝜑
2 oprabbid.2 ⊢ Ⅎ 𝑦 𝜑
3 oprabbid.3 ⊢ Ⅎ 𝑧 𝜑
4 oprabbid.4 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
5 4 anbi2d ⊢ ( 𝜑 → ( ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
6 3 5 exbid ⊢ ( 𝜑 → ( ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
7 2 6 exbid ⊢ ( 𝜑 → ( ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
8 1 7 exbid ⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) ) )
9 8 abbidv ⊢ ( 𝜑 → { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) } )
10 df-oprab ⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜓 ) }
11 df-oprab ⊢ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } = { 𝑤 ∣ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝑤 = ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∧ 𝜒 ) }
12 9 10 11 3eqtr4g ⊢ ( 𝜑 → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } = { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜒 } )