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

Theorem cbveu 1921
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
cbveu.1 yφ
cbveu.2 xψ
cbveu.3 (x = y → (φψ))
Assertion
Ref Expression
cbveu (∃!xφ∃!yψ)

Proof of Theorem cbveu
StepHypRef Expression
1 cbveu.1 . . 3 yφ
21sb8eu 1910 . 2 (∃!xφ∃!y[y / x]φ)
3 cbveu.2 . . . 4 xψ
4 cbveu.3 . . . 4 (x = y → (φψ))
53, 4sbie 1671 . . 3 ([y / x]φψ)
65eubii 1906 . 2 (∃!y[y / x]φ∃!yψ)
72, 6bitri 173 1 (∃!xφ∃!yψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wnf 1346  [wsb 1642  ∃!weu 1897
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
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900
This theorem is referenced by:  cbvmo  1937  cbvreu  2525  cbvreucsf  2904  tz6.12f  5145  f1ompt  5263
  Copyright terms: Public domain W3C validator