Theorem ssopab2dv 4716

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 1632	. 2 ⊢ (φ → ∀x∀y(ψ → χ))
3		ssopab2 4713	. 2 ⊢ (∀x∀y(ψ → χ) → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
4	2, 3	syl 15	1 ⊢ (φ → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})

Syntax hints:   → wi 4  ∀wal 1540   ⊆ wss 3258  {copab 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624

This theorem is referenced by:  xpss12  4856  coss1  4873  coss2  4874  cnvss  4886