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Metamath Proof Explorer

Theorem opabbidv

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995)

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

Proof

Step Hyp Ref Expression
1 opabbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 1 anbi2d ⊢ ( 𝜑 → ( ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜒 ) ) )
3 2 2exbidv ⊢ ( 𝜑 → ( ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜒 ) ) )
4 3 abbidv ⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜒 ) } )
5 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜓 ) }
6 df-opab ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜒 } = { 𝑧 ∣ ∃ 𝑥 ∃ 𝑦 ( 𝑧 = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜒 ) }
7 4 5 6 3eqtr4g ⊢ ( 𝜑 → { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜒 } )