Theorem dtruALT2

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

	Ref	Expression
Assertion	dtruALT2	⊢ ¬ ∀ 𝑥 𝑥 = 𝑦

Proof

Step	Hyp	Ref	Expression
1		elALT2	⊢ ∃ 𝑤 𝑥 ∈ 𝑤
2		ax-nul	⊢ ∃ 𝑧 ∀ 𝑥 ¬ 𝑥 ∈ 𝑧
3		elequ1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 ∈ 𝑧 ↔ 𝑤 ∈ 𝑧 ) )
4	3	notbid	⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑤 ∈ 𝑧 ) )
5	4	spw	⊢ ( ∀ 𝑥 ¬ 𝑥 ∈ 𝑧 → ¬ 𝑥 ∈ 𝑧 )
6	2 5	eximii	⊢ ∃ 𝑧 ¬ 𝑥 ∈ 𝑧
7		exdistrv	⊢ ( ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) ↔ ( ∃ 𝑤 𝑥 ∈ 𝑤 ∧ ∃ 𝑧 ¬ 𝑥 ∈ 𝑧 ) )
8	1 6 7	mpbir2an	⊢ ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 )
9		ax9v2	⊢ ( 𝑤 = 𝑧 → ( 𝑥 ∈ 𝑤 → 𝑥 ∈ 𝑧 ) )
10	9	com12	⊢ ( 𝑥 ∈ 𝑤 → ( 𝑤 = 𝑧 → 𝑥 ∈ 𝑧 ) )
11	10	con3dimp	⊢ ( ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) → ¬ 𝑤 = 𝑧 )
12	11	2eximi	⊢ ( ∃ 𝑤 ∃ 𝑧 ( 𝑥 ∈ 𝑤 ∧ ¬ 𝑥 ∈ 𝑧 ) → ∃ 𝑤 ∃ 𝑧 ¬ 𝑤 = 𝑧 )
13		equequ2	⊢ ( 𝑧 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑤 = 𝑦 ) )
14	13	notbid	⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦 ) )
15		ax7v1	⊢ ( 𝑥 = 𝑤 → ( 𝑥 = 𝑦 → 𝑤 = 𝑦 ) )
16	15	con3d	⊢ ( 𝑥 = 𝑤 → ( ¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦 ) )
17	16	spimevw	⊢ ( ¬ 𝑤 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 )
18	14 17	biimtrdi	⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) )
19		ax7v1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑦 → 𝑧 = 𝑦 ) )
20	19	con3d	⊢ ( 𝑥 = 𝑧 → ( ¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦 ) )
21	20	spimevw	⊢ ( ¬ 𝑧 = 𝑦 → ∃ 𝑥 ¬ 𝑥 = 𝑦 )
22	21	a1d	⊢ ( ¬ 𝑧 = 𝑦 → ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 ) )
23	18 22	pm2.61i	⊢ ( ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 )
24	23	exlimivv	⊢ ( ∃ 𝑤 ∃ 𝑧 ¬ 𝑤 = 𝑧 → ∃ 𝑥 ¬ 𝑥 = 𝑦 )
25	8 12 24	mp2b	⊢ ∃ 𝑥 ¬ 𝑥 = 𝑦
26		exnal	⊢ ( ∃ 𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀ 𝑥 𝑥 = 𝑦 )
27	25 26	mpbi	⊢ ¬ ∀ 𝑥 𝑥 = 𝑦