Theorem wl-clelv2-just 33868

Description: Show that the definition df-wl-clelv2 33869 is conservative. (Contributed by Wolf Lammen, 27-Nov-2021.)

Assertion

Ref	Expression
wl-clelv2-just	⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴))

Distinct variable group:   𝑥,𝑢,𝐴

Proof of Theorem wl-clelv2-just

Step	Hyp	Ref	Expression
1		ax-wl-8cl 33866	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 → 𝑥 ∈ 𝐴))
2		ax-wl-8cl 33866	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
3	2	equcoms 2119	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
4	1, 3	impbid 204	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	equsalvw 2103	. 2 ⊢ (∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
6	5	bicomi 216	1 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651   ∈ wcel-wl 33862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-wl-8cl 33866

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876

This theorem is referenced by: (None)