Theorem ssopab2dv 5432

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 → (𝜓 → 𝜒))

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

Step	Hyp	Ref	Expression
1		ssopab2dv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimivv 1936	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒))
3		ssopab2 5427	. 2 ⊢ (∀𝑥∀𝑦(𝜓 → 𝜒) → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})
4	2, 3	syl 17	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜒})

Syntax hints:   → wi 4  ∀wal 1541   ⊆ wss 3866  {copab 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-ss 3883  df-opab 5116

This theorem is referenced by:  xpss12  5566  coss1  5724  coss2  5725  cnvss  5741  aceq3lem  9734  coss12d  14535  shftfval  14633  sslm  22196  ulmval  25272  mptssALT  30732  fpwrelmap  30788  cossss  36285  dicssdvh  38937  rfovcnvf1od  41289