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

Theorem ltdfpr 6489
Description: More convenient form of df-iltp 6453. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltdfpr ((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
Distinct variable groups:   A,𝑞   B,𝑞

Proof of Theorem ltdfpr
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3756 . . 3 (A<P B ↔ ⟨A, B <P )
2 df-iltp 6453 . . . 4 <P = {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))}
32eleq2i 2101 . . 3 (⟨A, B <P ↔ ⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))})
41, 3bitri 173 . 2 (A<P B ↔ ⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))})
5 simpl 102 . . . . . . 7 ((x = A y = B) → x = A)
65fveq2d 5125 . . . . . 6 ((x = A y = B) → (2ndx) = (2ndA))
76eleq2d 2104 . . . . 5 ((x = A y = B) → (𝑞 (2ndx) ↔ 𝑞 (2ndA)))
8 simpr 103 . . . . . . 7 ((x = A y = B) → y = B)
98fveq2d 5125 . . . . . 6 ((x = A y = B) → (1sty) = (1stB))
109eleq2d 2104 . . . . 5 ((x = A y = B) → (𝑞 (1sty) ↔ 𝑞 (1stB)))
117, 10anbi12d 442 . . . 4 ((x = A y = B) → ((𝑞 (2ndx) 𝑞 (1sty)) ↔ (𝑞 (2ndA) 𝑞 (1stB))))
1211rexbidv 2321 . . 3 ((x = A y = B) → (𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)) ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
1312opelopab2a 3993 . 2 ((A P B P) → (⟨A, B {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))} ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
144, 13syl5bb 181 1 ((A P B P) → (A<P B𝑞 Q (𝑞 (2ndA) 𝑞 (1stB))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wrex 2301  cop 3370   class class class wbr 3755  {copab 3808  cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275  <P cltp 6279
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-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-iota 4810  df-fv 4853  df-iltp 6453
This theorem is referenced by:  nqprl  6533  ltprordil  6565  ltpopr  6569  ltsopr  6570  ltaddpr  6571  ltexprlemm  6574  ltexprlemopu  6577  ltexprlemru  6586  aptiprleml  6611  aptiprlemu  6612  archpr  6615  cauappcvgprlem2  6632  caucvgprlem2  6651
  Copyright terms: Public domain W3C validator