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Theorem hbequid 1403
 Description: Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x 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 (x = xy x = x)

Proof of Theorem hbequid
StepHypRef Expression
1 ax12or 1400 . 2 (y y = x (y y = x y(x = xy x = x)))
2 ax-8 1392 . . . . . 6 (y = x → (y = xx = x))
32pm2.43i 43 . . . . 5 (y = xx = x)
43alimi 1341 . . . 4 (y y = xy x = x)
54a1d 22 . . 3 (y y = x → (x = xy x = x))
6 ax-4 1397 . . . 4 (y(x = xy x = x) → (x = xy x = x))
75, 6jaoi 635 . . 3 ((y y = x y(x = xy x = x)) → (x = xy x = x))
85, 7jaoi 635 . 2 ((y y = x (y y = x y(x = xy x = x))) → (x = xy x = x))
91, 8ax-mp 7 1 (x = xy x = x)
 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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