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Theorem ssopab2bw 5505
Description: Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5507 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Dec-1996.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.)
Assertion
Ref Expression
ssopab2bw ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem ssopab2bw
StepHypRef Expression
1 nfopab1 5176 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab1 5176 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜓}
31, 2nfss 3937 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
4 nfopab2 5177 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
5 nfopab2 5177 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜓}
64, 5nfss 3937 . . . 4 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}
7 ssel 3938 . . . . 5 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓}))
8 opabidw 5482 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
9 opabidw 5482 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜓)
107, 8, 93imtr3g 295 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → (𝜑𝜓))
116, 10alrimi 2207 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑦(𝜑𝜓))
123, 11alrimi 2207 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → ∀𝑥𝑦(𝜑𝜓))
13 ssopab2 5504 . 2 (∀𝑥𝑦(𝜑𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
1412, 13impbii 208 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  wss 3911  cop 4593  {copab 5168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5169
This theorem is referenced by:  eqopab2bw  5506  dffun2OLDOLD  6509  marypha2lem3  9374  cvmlift2lem12  33911  cossssid2  36933
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