Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8eh Structured version   GIF version

Theorem sb8eh 1732
 Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8eh.1 (φyφ)
Assertion
Ref Expression
sb8eh (xφy[y / x]φ)

Proof of Theorem sb8eh
StepHypRef Expression
1 sb8eh.1 . 2 (φyφ)
21hbsb3 1686 . 2 ([y / x]φx[y / x]φ)
3 sbequ12 1651 . 2 (x = y → (φ ↔ [y / x]φ))
41, 2, 3cbvexh 1635 1 (xφy[y / x]φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642 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 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  exsb  1881
 Copyright terms: Public domain W3C validator