Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rmoeq1f Structured version   GIF version

Theorem rmoeq1f 2482
 Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypotheses
Ref Expression
raleq1f.1 xA
raleq1f.2 xB
Assertion
Ref Expression
rmoeq1f (A = B → (∃*x A φ∃*x B φ))

Proof of Theorem rmoeq1f
StepHypRef Expression
1 raleq1f.1 . . . 4 xA
2 raleq1f.2 . . . 4 xB
31, 2nfeq 2167 . . 3 x A = B
4 eleq2 2083 . . . 4 (A = B → (x Ax B))
54anbi1d 441 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5mobid 1917 . 2 (A = B → (∃*x(x A φ) ↔ ∃*x(x B φ)))
7 df-rmo 2292 . 2 (∃*x A φ∃*x(x A φ))
8 df-rmo 2292 . 2 (∃*x B φ∃*x(x B φ))
96, 7, 83bitr4g 212 1 (A = B → (∃*x A φ∃*x B φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∃*wmo 1883  Ⅎwnfc 2147  ∃*wrmo 2287 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rmo 2292 This theorem is referenced by:  rmoeq1  2486
 Copyright terms: Public domain W3C validator