Theorem wl-clelv2-just 33379

Description: Show that the definition df-wl-clelv2 33380 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 33377	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 → 𝑥 ∈ 𝐴))
2		ax-wl-8cl 33377	. . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
3	2	equcoms 1947	. . . 4 ⊢ (𝑢 = 𝑥 → (𝑥 ∈ 𝐴 → 𝑢 ∈ 𝐴))
4	1, 3	impbid 202	. . 3 ⊢ (𝑢 = 𝑥 → (𝑢 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	equsalvw 1931	. 2 ⊢ (∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
6	5	bicomi 214	1 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   ∈ 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)