Theorem ssoprab2b

Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by FL, 6-Nov-2013) (Proof shortened by Mario Carneiro, 11-Dec-2016) (New usage is discouraged.)

	Ref	Expression
Assertion	ssoprab2b	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfoprab1	⊢ Ⅎ 𝑥 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
2		nfoprab1	⊢ Ⅎ 𝑥 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
3	1 2	nfss	⊢ Ⅎ 𝑥 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
4		nfoprab2	⊢ Ⅎ 𝑦 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
5		nfoprab2	⊢ Ⅎ 𝑦 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
6	4 5	nfss	⊢ Ⅎ 𝑦 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
7		nfoprab3	⊢ Ⅎ 𝑧 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 }
8		nfoprab3	⊢ Ⅎ 𝑧 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
9	7 8	nfss	⊢ Ⅎ 𝑧 { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 }
10		ssel	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } → ( ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } → ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } ) )
11		oprabid	⊢ ( ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ↔ 𝜑 )
12		oprabid	⊢ ( ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∈ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } ↔ 𝜓 )
13	10 11 12	3imtr3g	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } → ( 𝜑 → 𝜓 ) )
14	9 13	alrimi	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } → ∀ 𝑧 ( 𝜑 → 𝜓 ) )
15	6 14	alrimi	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } → ∀ 𝑦 ∀ 𝑧 ( 𝜑 → 𝜓 ) )
16	3 15	alrimi	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } → ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 → 𝜓 ) )
17		ssoprab2	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 → 𝜓 ) → { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } )
18	16 17	impbii	⊢ ( { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜑 } ⊆ { ⟨ ⟨ 𝑥 , 𝑦 ⟩ , 𝑧 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( 𝜑 → 𝜓 ) )