Theorem clel2g 3556

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 2177. (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 2812	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		biimt 364	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵)))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵)))
4		19.23v 1950	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
5		eleq1 2818	. . . . . 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 1827	. . 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 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809

This theorem is referenced by:  clel2  3558  snssg  4684