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Theorem hbequid 33106
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-c10 33083.) (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 = 𝑥 → ∀𝑦 𝑥 = 𝑥)

Proof of Theorem hbequid
StepHypRef Expression
1 ax-c9 33087 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)))
2 ax7 1893 . . . . 5 (𝑦 = 𝑥 → (𝑦 = 𝑥𝑥 = 𝑥))
32pm2.43i 49 . . . 4 (𝑦 = 𝑥𝑥 = 𝑥)
43alimi 1715 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
54a1d 25 . 2 (∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥))
61, 5, 5pm2.61ii 175 1 (𝑥 = 𝑥 → ∀𝑦 𝑥 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-c9 33087
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  nfequid-o  33107  equidq  33121
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