Theorem in-ax8 37013

Description: A proof of ax-8 2146 that does not rely on ax-8 2146. It employs df-in 4091 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2154. 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 2048	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 → 𝑦 = 𝑤))
2		ax12v2 2214	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))))
3	2	imp 412	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑤 ∧ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)))
4		sb6 2121	. . . . . . . . . . . . 13 ⊢ ([𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) ↔ ∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)))
5		df-in 4091	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ 𝑡) = {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)}
6		df-in 4091	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∩ 𝑡) = {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)}
7	5, 6	eqtr3i 2785	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} = {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)}
8		dfcleq 2753	. . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . 15 ⊢ ∀𝑤(𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ 𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)})
10	9	spi 2220	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ 𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)})
11		df-clab 2739	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑥 ∣ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)} ↔ [𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))
12		df-clab 2739	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {𝑦 ∣ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)} ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
13	10, 11, 12	3bitr3i 304	. . . . . . . . . . . . 13 ⊢ ([𝑤 / 𝑥](𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
14	4, 13	bitr3i 280	. . . . . . . . . . . 12 ⊢ (∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) ↔ [𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
15		sb6 2121	. . . . . . . . . . . 12 ⊢ ([𝑤 / 𝑦](𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡) ↔ ∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
16	14, 15	sylbb 222	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 = 𝑤 → (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → ∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
17		sp 2219	. . . . . . . . . . 11 ⊢ (∀𝑦(𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
18	3, 16, 17	3syl 19	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑤 ∧ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡)) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
19	18	ex 418	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 = 𝑤 → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
20	19	com23 87	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑦 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
21	1, 20	sylcom 31	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑤 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
22	21	com12 33	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
23	22	equcoms 2052	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))))
24		ax6ev 2001	. . . . 5 ⊢ ∃𝑤 𝑤 = 𝑥
25	23, 24	exlimiiv 1963	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡) → (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡)))
26		pm4.24 574	. . . 4 ⊢ (𝑥 ∈ 𝑡 ↔ (𝑥 ∈ 𝑡 ∧ 𝑥 ∈ 𝑡))
27		pm4.24 574	. . . 4 ⊢ (𝑦 ∈ 𝑡 ↔ (𝑦 ∈ 𝑡 ∧ 𝑦 ∈ 𝑡))
28	25, 26, 27	3imtr4g 299	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑡 → 𝑦 ∈ 𝑡))
29		ax9 2158	. . . . 5 ⊢ (𝑧 = 𝑡 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑡))
30	29	equcoms 2052	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑡))
31		ax9 2158	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑦 ∈ 𝑡 → 𝑦 ∈ 𝑧))
32	30, 31	imim12d 82	. . 3 ⊢ (𝑡 = 𝑧 → ((𝑥 ∈ 𝑡 → 𝑦 ∈ 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
33	28, 32	syl5 35	. 2 ⊢ (𝑡 = 𝑧 → (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)))
34		ax6ev 2001	. 2 ⊢ ∃𝑡 𝑡 = 𝑧
35	33, 34	exlimiiv 1963	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  [wsb 2098   ∈ wcel 2144  {cab 2738   ∩ cin 4089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-in 4091

This theorem is used by: (None)