Theorem dtruALT2 5342

Description: Alternate proof of dtru 5419 using ax-pow 5337 instead of ax-pr 5405. See dtruALT 5360 for another proof using ax-pow 5337 instead of ax-pr 5405. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2410. (Revised by BJ, 31-May-2019.) Avoid ax-12 2219. (Revised by Rohan Ridenour, 9-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
dtruALT2	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elALT2 5341	. . . 4 ⊢ ∃𝑤 𝑥 ∈ 𝑤
2		ax-nul 5271	. . . . 5 ⊢ ∃𝑧∀𝑥 ¬ 𝑥 ∈ 𝑧
3		elequ1 2156	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧))
4	3	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝑤 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑤 ∈ 𝑧))
5	4	spw 2061	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧)
6	2, 5	eximii 1864	. . . 4 ⊢ ∃𝑧 ¬ 𝑥 ∈ 𝑧
7		exdistrv 1982	. . . 4 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) ↔ (∃𝑤 𝑥 ∈ 𝑤 ∧ ∃𝑧 ¬ 𝑥 ∈ 𝑧))
8	1, 6, 7	mpbir2an 723	. . 3 ⊢ ∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧)
9		ax9v2 2162	. . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧))
10	9	com12 33	. . . . 5 ⊢ (𝑥 ∈ 𝑤 → (𝑤 = 𝑧 → 𝑥 ∈ 𝑧))
11	10	con3dimp 413	. . . 4 ⊢ ((𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ¬ 𝑤 = 𝑧)
12	11	2eximi 1863	. . 3 ⊢ (∃𝑤∃𝑧(𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧) → ∃𝑤∃𝑧 ¬ 𝑤 = 𝑧)
13		equequ2 2053	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑤 = 𝑧 ↔ 𝑤 = 𝑦))
14	13	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
15		ax7v1 2037	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → 𝑤 = 𝑦))
16	15	con3d 153	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
17	16	spimevw 2012	. . . . . 6 ⊢ (¬ 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
18	14, 17	biimtrdi 256	. . . . 5 ⊢ (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19		ax7v1 2037	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
20	19	con3d 153	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
21	20	spimevw 2012	. . . . . 6 ⊢ (¬ 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
22	21	a1d 26	. . . . 5 ⊢ (¬ 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
23	18, 22	pm2.61i 184	. . . 4 ⊢ (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
24	23	exlimivv 1959	. . 3 ⊢ (∃𝑤∃𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
25	8, 12, 24	mp2b 10	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
26		exnal 1854	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
27	25, 26	mpbi 233	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-nul 5271  ax-pow 5337

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  dtrucor  5343  dvdemo1  5345  nfnid  5347  axc16b  5361  eunex  5362  brprcneuALT  6873