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Theorem ssopab2dv 5556
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 1928 . 2 (𝜑 → ∀𝑥𝑦(𝜓𝜒))
3 ssopab2 5551 . 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 1538  wss 3951  {copab 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ss 3968  df-opab 5206
This theorem is referenced by:  xpss12  5700  coss1  5866  coss2  5867  cnvss  5883  aceq3lem  10160  coss12d  15011  shftfval  15109  sslm  23307  ulmval  26423  mptssALT  32685  fpwrelmap  32744  cossss  38426  dicssdvh  41188  rfovcnvf1od  44017
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