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

Theorem rabeqf 2544
Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
Hypotheses
Ref Expression
rabeqf.1 xA
rabeqf.2 xB
Assertion
Ref Expression
rabeqf (A = B → {x Aφ} = {x Bφ})

Proof of Theorem rabeqf
StepHypRef Expression
1 rabeqf.1 . . . 4 xA
2 rabeqf.2 . . . 4 xB
31, 2nfeq 2182 . . 3 x A = B
4 eleq2 2098 . . . 4 (A = B → (x Ax B))
54anbi1d 438 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5abbid 2151 . 2 (A = B → {x ∣ (x A φ)} = {x ∣ (x B φ)})
7 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
8 df-rab 2309 . 2 {x Bφ} = {x ∣ (x B φ)}
96, 7, 83eqtr4g 2094 1 (A = B → {x Aφ} = {x Bφ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {cab 2023  wnfc 2162  {crab 2304
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309
This theorem is referenced by:  rabeq  2545
  Copyright terms: Public domain W3C validator