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Theorem hbae 1729
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 1519 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax10o 1726 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32alequcoms 1527 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
4 ax10o 1726 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
54pm2.43i 49 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
6 ax10o 1726 . . . . . . . 8 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
75, 6syl5 32 . . . . . . 7 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
87alequcoms 1527 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9 ax-4 1521 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
109imim1i 60 . . . . . . 7 ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1110sps 1548 . . . . . 6 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
128, 11jaoi 717 . . . . 5 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
133, 12jaoi 717 . . . 4 ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
141, 13ax-mp 5 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
1514a5i 1554 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
16 ax-7 1459 . 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 709  wal 1362   = wceq 1364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  nfae  1730  hbaes  1731  hbnae  1732  dral1  1741  dral2  1742  drex2  1743  drex1  1809  aev  1823  sbcomxyyz  1984  exists1  2134
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