Theorem clel2 2907

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
clel2	⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem clel2

Step	Hyp	Ref	Expression
1		clel2.1	. . 3 ⊢ 𝐴 ∈ V
2		eleq1 2269	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ceqsalv 2803	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ 𝐴 ∈ 𝐵)
4	3	bicomi 132	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1371   = wceq 1373   ∈ wcel 2177  Vcvv 2773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-v 2775

This theorem is referenced by:  snssOLD  3761