Theorem clel2g 3619

Description: Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022.) Avoid ax-12 2213. (Revised by BJ, 1-Sep-2024.)

Assertion

Ref	Expression
clel2g	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel2g

Step	Hyp	Ref	Expression
1		elisset 2845	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		biimt 363	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵)))
3	1, 2	syl 18	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵)))
4		19.23v 1972	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
5		eleq1 2851	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6	5	bicomd 226	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵))
7	6	pm5.74i 274	. . . 4 ⊢ ((𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
8	7	albii 1849	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
9	4, 8	bitr3i 280	. 2 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
10	3, 9	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838

This theorem is referenced by:  clel2  3620