Theorem axbndOLD 2792

Description: Obsolete version of axbnd 2791 as of 24-Apr-2023. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axbndOLD	⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Proof of Theorem axbndOLD

Step	Hyp	Ref	Expression
1		nfnae 2452	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑥
2		nfnae 2452	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1896	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		nfnae 2452	. . . . . . 7 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
5		nfnae 2452	. . . . . . 7 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6	4, 5	nfan 1896	. . . . . 6 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
7		axc9 2396	. . . . . . 7 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
8	7	imp 409	. . . . . 6 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
9	6, 8	alrimi 2208	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
10	3, 9	alrimi 2208	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
11	10	ex 415	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
12	11	orrd 859	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
13	12	orri 858	1 ⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2156  ax-12 2172  ax-13 2386

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781

This theorem is referenced by: (None)