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Metamath Proof Explorer

Theorem opabbid

Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 21-Feb-2004) (Proof shortened by Andrew Salmon, 9-Jul-2011)

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

Proof

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