ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssopab2dv Structured version   GIF version

Theorem ssopab2dv 4006
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 1752 . 2 (φxy(ψχ))
3 ssopab2 4003 . 2 (xy(ψχ) → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
42, 3syl 14 1 (φ → {⟨x, y⟩ ∣ ψ} ⊆ {⟨x, y⟩ ∣ χ})
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wss 2911  {copab 3808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810
This theorem is referenced by:  xpss12  4388  coss1  4434  coss2  4435  cnvss  4451
  Copyright terms: Public domain W3C validator