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Metamath Proof Explorer

Theorem ssopab2bw

Description: Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 27-Dec-1996) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

Ref Expression
Assertion ssopab2bw ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )

Proof

Step Hyp Ref Expression
1 nfopab1 ⊢ Ⅎ 𝑥 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
2 nfopab1 ⊢ Ⅎ 𝑥 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 }
3 1 2 nfss ⊢ Ⅎ 𝑥 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 }
4 nfopab2 ⊢ Ⅎ 𝑦 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
5 nfopab2 ⊢ Ⅎ 𝑦 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 }
6 4 5 nfss ⊢ Ⅎ 𝑦 { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 }
7 ssel ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } → ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ) )
8 opabidw ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜑 )
9 opabidw ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ 𝜓 )
10 7 8 9 3imtr3g ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → ( 𝜑 → 𝜓 ) )
11 6 10 alrimi ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → ∀ 𝑦 ( 𝜑 → 𝜓 ) )
12 3 11 alrimi ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } → ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )
13 ssopab2 ⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) → { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } )
14 12 13 impbii ⊢ ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜓 } ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 → 𝜓 ) )