Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbequid Unicode version

Theorem hbequid 1687
 Description: Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1533, ax-8 1623, ax-12o 1664, and ax-gen 1536. This shows that this can be proved without ax-9 1684, even though the theorem equid 1818 cannot be. A shorter proof using ax-9 1684 is obtainable from equid 1818 and hbth 1557.) Remark added 2-Dec-2015 NM: This proof does implicitly use ax-9v 1632, which is used for the derivation of ax12o 1663, unless we consider ax-12o 1664 the starting axiom rather than ax-12 1633. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.)
Assertion
Ref Expression
hbequid

Proof of Theorem hbequid
StepHypRef Expression
1 ax-12o 1664 . 2
2 ax-8 1623 . . . . 5
32pm2.43i 45 . . . 4
43alimi 1546 . . 3
54a1d 24 . 2
61, 5, 5pm2.61ii 159 1
 Colors of variables: wff set class Syntax hints:   wi 6  wal 1532 This theorem is referenced by:  nfequid  1688  equidq  1689 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-8 1623  ax-12o 1664
 Copyright terms: Public domain W3C validator