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Theorem hbequid 2160
 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 does not use ax-9o 2138.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbequid (x = xy x = x)

Proof of Theorem hbequid
StepHypRef Expression
1 ax-12o 2142 . 2 y y = x → (¬ y y = x → (x = xy x = x)))
2 ax-8 1675 . . . . 5 (y = x → (y = xx = x))
32pm2.43i 43 . . . 4 (y = xx = x)
43alimi 1559 . . 3 (y y = xy x = x)
54a1d 22 . 2 (y y = x → (x = xy x = x))
61, 5, 5pm2.61ii 157 1 (x = xy x = x)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-8 1675  ax-12o 2142 This theorem is referenced by:  nfequid-o  2161  equidq  2175
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