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Theorem hbequid 1403
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 1333, ax-8 1392, ax-12 1399, and ax-gen 1335. This shows that this can be proved without ax-9 1421, even though the theorem equid 1586 cannot be. A shorter proof using ax-9 1421 is obtainable from equid 1586 and hbth 1349.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.)
Assertion
Ref Expression
hbequid

Proof of Theorem hbequid
StepHypRef Expression
1 ax12or 1400 . 2
2 ax-8 1392 . . . . . 6
32pm2.43i 43 . . . . 5
43alimi 1341 . . . 4
54a1d 22 . . 3
6 ax-4 1397 . . . 4
75, 6jaoi 635 . . 3
85, 7jaoi 635 . 2
91, 8ax-mp 7 1
Colors of variables: wff set class
Syntax hints:   wi 4   wo 628  wal 1240   wceq 1242
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-gen 1335  ax-8 1392  ax-i12 1395  ax-4 1397
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  equveli  1639
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