Theorem clel2gOLD 3582

Description: Obsolete version of clel2g 3581 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
clel2gOLD	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel2gOLD

Step	Hyp	Ref	Expression
1		nfv 1918	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
2		eleq1 2826	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ceqsalg 3454	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ 𝐴 ∈ 𝐵))
4	3	bicomd 222	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817

This theorem is referenced by: (None)