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Theorem hbae 2314
Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
Assertion
Ref Expression
hbae (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Proof of Theorem hbae
StepHypRef Expression
1 sp 2051 . . . . 5 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 axc9 2301 . . . . 5 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
31, 2syl7 74 . . . 4 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
4 axc11r 2186 . . . 4 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
5 axc11 2313 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
65pm2.43i 52 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
7 axc11r 2186 . . . . 5 (∀𝑧 𝑧 = 𝑦 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
86, 7syl5 34 . . . 4 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
93, 4, 8pm2.61ii 177 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
109axc4i 2127 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
11 ax-11 2031 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
1210, 11syl 17 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  nfae  2315  hbnae  2316  aevALTOLD  2320  drex2  2327  ax6e2eq  38294
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