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Theorem ssopab2dv 4168
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 1829 . 2 (𝜑 → ∀𝑥𝑦(𝜓𝜒))
3 ssopab2 4165 . 2 (∀𝑥𝑦(𝜓𝜒) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
42, 3syl 14 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312  wss 3039  {copab 3956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-in 3045  df-ss 3052  df-opab 3958
This theorem is referenced by:  xpss12  4614  coss1  4662  coss2  4663  cnvss  4680  shftfvalg  10541  shftfval  10544  sslm  12322
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