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Theorem ssopab2dv 5165
Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 19-Jan-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypothesis
Ref Expression
ssopab2dv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ssopab2dv (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem ssopab2dv
StepHypRef Expression
1 ssopab2dv.1 . . 3 (𝜑 → (𝜓𝜒))
21alrimivv 2023 . 2 (𝜑 → ∀𝑥𝑦(𝜓𝜒))
3 ssopab2 5162 . 2 (∀𝑥𝑦(𝜓𝜒) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
42, 3syl 17 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wss 3732  {copab 4871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-in 3739  df-ss 3746  df-opab 4872
This theorem is referenced by:  xpss12  5292  coss1  5446  coss2  5447  cnvss  5463  aceq3lem  9194  coss12d  14000  shftfval  14097  sslm  21383  ulmval  24425  mptssALT  29858  fpwrelmap  29892  cossss  34541  dicssdvh  37074  rfovcnvf1od  38904
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