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

Theorem opabbidv 3814
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
opabbidv.1 (φ → (ψχ))
Assertion
Ref Expression
opabbidv (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 opabbidv
StepHypRef Expression
1 nfv 1418 . 2 xφ
2 nfv 1418 . 2 yφ
3 opabbidv.1 . 2 (φ → (ψχ))
41, 2, 3opabbid 3813 1 (φ → {⟨x, y⟩ ∣ ψ} = {⟨x, y⟩ ∣ χ})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  {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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-opab 3810
This theorem is referenced by:  opabbii  3815  csbopabg  3826  xpeq1  4302  xpeq2  4303  opabbi2dv  4428  csbcnvg  4462  resopab2  4598  cores  4767  xpcom  4807  dffn5im  5162  f1oiso2  5409  f1ocnvd  5644  ofreq  5657  sprmpt2  5798
  Copyright terms: Public domain W3C validator