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Theorem ssopab2dv 5403
 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 1929 . 2 (𝜑 → ∀𝑥𝑦(𝜓𝜒))
3 ssopab2 5398 . 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 1536   ⊆ wss 3881  {copab 5092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-opab 5093 This theorem is referenced by:  xpss12  5534  coss1  5690  coss2  5691  cnvss  5707  aceq3lem  9533  coss12d  14325  shftfval  14423  sslm  21911  ulmval  24982  mptssALT  30445  fpwrelmap  30502  cossss  35846  dicssdvh  38498  rfovcnvf1od  40720
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