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Theorem ssopab2b 5401
Description: Equivalence of ordered pair abstraction subclass and implication. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker ssopab2bw 5399 when possible. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.)
Assertion
Ref Expression
ssopab2b ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))

Proof of Theorem ssopab2b
StepHypRef Expression
1 nfopab1 5096 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 5096 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜓}
31, 2nfss 3867 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
4 nfopab2 5097 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 5097 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜓}
64, 5nfss 3867 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
7 ssel 3868 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
8 opabid 5378 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
9 opabid 5378 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜓)
107, 8, 93imtr3g 298 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (𝜑𝜓))
116, 10alrimi 2214 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑦(𝜑𝜓))
123, 11alrimi 2214 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑥𝑦(𝜑𝜓))
13 ssopab2 5398 . 2 (∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
1412, 13impbii 212 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540  wcel 2113  wss 3841  cop 4519  {copab 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-13 2371  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-opab 5090
This theorem is referenced by:  eqopab2b  5404
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