Theorem clel4g 3586

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2173. (Revised by BJ, 1-Sep-2024.)

Assertion

Ref	Expression
clel4g	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem clel4g

Step	Hyp	Ref	Expression
1		elisset 2820	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵)
2		biimt 360	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵)))
3	1, 2	syl 17	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵)))
4		19.23v 1946	. . 3 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 → 𝐴 ∈ 𝐵))
5	3, 4	bitr4di 288	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵)))
6		eleq2 2827	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
7	6	bicomd 222	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥))
8	7	pm5.74i 270	. . 3 ⊢ ((𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
9	8	albii 1823	. 2 ⊢ (∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥))
10	5, 9	bitrdi 286	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ 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-ext 2709

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

This theorem is referenced by:  clel4  3587  intprg  4909