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Theorem hbae 1653
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 1448 . . . 4 (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2 ax10o 1650 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
32alequcoms 1454 . . . . 5 (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
4 ax10o 1650 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
54pm2.43i 48 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
6 ax10o 1650 . . . . . . . 8 (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
75, 6syl5 32 . . . . . . 7 (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
87alequcoms 1454 . . . . . 6 (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9 ax-4 1445 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
109imim1i 59 . . . . . . 7 ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
1110sps 1475 . . . . . 6 (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
128, 11jaoi 671 . . . . 5 ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
133, 12jaoi 671 . . . 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 1480 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑥𝑧 𝑥 = 𝑦)
16 ax-7 1382 . 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 664  wal 1287   = wceq 1289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  nfae  1654  hbaes  1655  hbnae  1656  dral1  1665  dral2  1666  drex2  1667  drex1  1726  aev  1740  sbcomxyyz  1894  exists1  2044
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