Theorem wl-ax8clv2 33381

Description: Axiom ax-wl-8cl 33377 carries over to our new definition df-wl-clelv2 33380. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-ax8clv2	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))

Proof of Theorem wl-ax8clv2

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

Step	Hyp	Ref	Expression
1		equvinv 1959	. 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑢(𝑢 = 𝑥 ∧ 𝑢 = 𝑦))
2		df-wl-clelv2 33380	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑡(𝑡 = 𝑥 → 𝑡 ∈ 𝐴))
3		equtrr 1949	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑡 = 𝑢 → 𝑡 = 𝑥))
4	3	imim1d 82	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑡 = 𝑥 → 𝑡 ∈ 𝐴) → (𝑡 = 𝑢 → 𝑡 ∈ 𝐴)))
5	4	alimdv 1845	. . . . 5 ⊢ (𝑢 = 𝑥 → (∀𝑡(𝑡 = 𝑥 → 𝑡 ∈ 𝐴) → ∀𝑡(𝑡 = 𝑢 → 𝑡 ∈ 𝐴)))
6	2, 5	syl5bi 232	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑡(𝑡 = 𝑢 → 𝑡 ∈ 𝐴)))
7		equeuclr 1950	. . . . . . 7 ⊢ (𝑢 = 𝑦 → (𝑡 = 𝑦 → 𝑡 = 𝑢))
8	7	imim1d 82	. . . . . 6 ⊢ (𝑢 = 𝑦 → ((𝑡 = 𝑢 → 𝑡 ∈ 𝐴) → (𝑡 = 𝑦 → 𝑡 ∈ 𝐴)))
9	8	alimdv 1845	. . . . 5 ⊢ (𝑢 = 𝑦 → (∀𝑡(𝑡 = 𝑢 → 𝑡 ∈ 𝐴) → ∀𝑡(𝑡 = 𝑦 → 𝑡 ∈ 𝐴)))
10		df-wl-clelv2 33380	. . . . 5 ⊢ (𝑦 ∈ 𝐴 ↔ ∀𝑡(𝑡 = 𝑦 → 𝑡 ∈ 𝐴))
11	9, 10	syl6ibr 242	. . . 4 ⊢ (𝑢 = 𝑦 → (∀𝑡(𝑡 = 𝑢 → 𝑡 ∈ 𝐴) → 𝑦 ∈ 𝐴))
12	6, 11	sylan9 689	. . 3 ⊢ ((𝑢 = 𝑥 ∧ 𝑢 = 𝑦) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
13	12	exlimiv 1858	. 2 ⊢ (∃𝑢(𝑢 = 𝑥 ∧ 𝑢 = 𝑦) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
14	1, 13	sylbi 207	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704   ∈ wcel-wl 33373   ∈ wcel2-wl 33375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-wl-clelv2 33380

This theorem is referenced by: (None)