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Theorem hbae 1606
 Description: All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
Assertion
Ref Expression
hbae (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Proof of Theorem hbae
StepHypRef Expression
1 ax12or 1403 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax10o 1603 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32alequcoms 1409 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
4 ax10o 1603 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
54pm2.43i 43 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
6 ax10o 1603 . . . . . . . 8 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
75, 6syl5 28 . . . . . . 7 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
87alequcoms 1409 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9 ax-4 1400 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
109imim1i 54 . . . . . . 7 ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1110sps 1430 . . . . . 6 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
128, 11jaoi 636 . . . . 5 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
133, 12jaoi 636 . . . 4 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
141, 13ax-mp 7 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1514a5i 1435 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
16 ax-7 1337 . 2 (∀𝑥𝑧 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
1715, 16syl 14 1 (∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629  ∀wal 1241   = wceq 1243 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  nfae  1607  hbaes  1608  hbnae  1609  dral1  1618  dral2  1619  drex2  1620  drex1  1679  aev  1693  sbcomxyyz  1846  exists1  1996
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