Theorem in-ax8 36203

Description: A proof of ax-8 2110 that does not rely on ax-8 2110. It employs df-in 3957 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2118. Since the nature of this result is unclear, usage of this theorem is discouraged, and this method should not be applied to eliminate axiom dependencies. (Contributed by GG, 1-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
in-ax8	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Proof of Theorem in-ax8

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

Step	Hyp	Ref	Expression
1		ax7 2015	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 → 𝑦 = 𝑤))
2		ax12v2 2179	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))))
3	2	imp 406	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)))
4		sb6 2085	. . . . . . . . . . . . 13 ⊢ ([𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) ↔ ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)))
5		df-in 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ 𝑡) = {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)}
6		df-in 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ 𝑡) = {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)}
7	5, 6	eqtr3i 2766	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} = {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)}
8		dfcleq 2729	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} = {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)} ↔ ∀𝑤(𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ 𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)}))
9	7, 8	mpbi 230	. . . . . . . . . . . . . . 15 ⊢ ∀𝑤(𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ 𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)})
10	9	spi 2184	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ 𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)})
11		df-clab 2714	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ [𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))
12		df-clab 2714	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)} ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
13	10, 11, 12	3bitr3i 301	. . . . . . . . . . . . 13 ⊢ ([𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
14	4, 13	bitr3i 277	. . . . . . . . . . . 12 ⊢ (∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
15		sb6 2085	. . . . . . . . . . . 12 ⊢ ([𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡) ↔ ∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
16	14, 15	sylbb 219	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → ∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
17		sp 2183	. . . . . . . . . . 11 ⊢ (∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
18	3, 16, 17	3syl 18	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑤 ∧ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
19	18	ex 412	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
20	19	com23 86	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
21	1, 20	sylcom 30	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
22	21	com12 32	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
23	22	equcoms 2019	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
24		ax6ev 1969	. . . . 5 ⊢ ∃𝑤 𝑤 = 𝑥
25	23, 24	exlimiiv 1931	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
26		pm4.24 563	. . . 4 ⊢ (𝑥 ∈ 𝑡 ↔ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))
27		pm4.24 563	. . . 4 ⊢ (𝑦 ∈ 𝑡 ↔ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
28	25, 26, 27	3imtr4g 296	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑡 → 𝑦 ∈ 𝑡))
29		ax9 2122	. . . . 5 ⊢ (𝑧 = 𝑡 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑡))
30	29	equcoms 2019	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑡))
31		ax9 2122	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑦 ∈ 𝑡 → 𝑦 ∈ 𝑧))
32	30, 31	imim12d 81	. . 3 ⊢ (𝑡 = 𝑧 → ((𝑥 ∈ 𝑡 → 𝑦 ∈ 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
33	28, 32	syl5 34	. 2 ⊢ (𝑡 = 𝑧 → (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
34		ax6ev 1969	. 2 ⊢ ∃𝑡 𝑡 = 𝑧
35	33, 34	exlimiiv 1931	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  [wsb 2064   ∈ wcel 2108  {cab 2713   ∩ cin 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-in 3957

This theorem is referenced by: (None)