Theorem wl-ax8clv1 33378

Description: Lifting the distinct variable constraint on 𝑥 and 𝑦 in ax-wl-8cl 33377. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-ax8clv1	⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem wl-ax8clv1

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinv 1959	. 2 ⊢ (𝑥 = 𝑦 ↔ ∃𝑢(𝑢 = 𝑥 ∧ 𝑢 = 𝑦))
2		ax-wl-8cl 33377	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
3	2	equcoms 1947	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
4		ax-wl-8cl 33377	. . . 4 ⊢ (𝑢 = 𝑦 → (𝑢 ∈ 𝐴 → 𝑦 ∈ 𝐴))
5	3, 4	sylan9 689	. . 3 ⊢ ((𝑢 = 𝑥 ∧ 𝑢 = 𝑦) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
6	5	exlimiv 1858	. 2 ⊢ (∃𝑢(𝑢 = 𝑥 ∧ 𝑢 = 𝑦) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))
7	1, 6	sylbi 207	1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel-wl 33373

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  ax-wl-8cl 33377

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

This theorem is referenced by: (None)