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 → (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 1928	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝜓 → 𝜒))
3		ssopab2 5551	. 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 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