Theorem hbae 1718

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 → ∀𝑧∀𝑥 𝑥 = 𝑦)

Proof of Theorem hbae

Step	Hyp	Ref	Expression
1		ax12or 1508	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
2		ax10o 1715	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
3	2	alequcoms 1516	. . . . 5 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
4		ax10o 1715	. . . . . . . . 9 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
5	4	pm2.43i 49	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
6		ax10o 1715	. . . . . . . 8 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑦 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7	5, 6	syl5 32	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
8	7	alequcoms 1516	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9		ax-4 1510	. . . . . . . 8 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
10	9	imim1i 60	. . . . . . 7 ⊢ ((𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11	10	sps 1537	. . . . . 6 ⊢ (∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
12	8, 11	jaoi 716	. . . . 5 ⊢ ((∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
13	3, 12	jaoi 716	. . . 4 ⊢ ((∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) → (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
14	1, 13	ax-mp 5	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
15	14	a5i 1543	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑧 𝑥 = 𝑦)
16		ax-7 1448	. 2 ⊢ (∀𝑥∀𝑧 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
17	15, 16	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∨ wo 708  ∀wal 1351   = wceq 1353

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfae  1719  hbaes  1720  hbnae  1721  dral1  1730  dral2  1731  drex2  1732  drex1  1798  aev  1812  sbcomxyyz  1972  exists1  2122