Theorem bj-issetwt 36841

Description: Closed form of bj-issetw 36842. (Contributed by BJ, 29-Apr-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-issetwt	⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem bj-issetwt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclel 2820	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}))
2	1	a1i 11	. 2 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑})))
3		vexwt 2722	. . . . 5 ⊢ (∀𝑥𝜑 → 𝑧 ∈ {𝑥 ∣ 𝜑})
4	3	biantrud 531	. . . 4 ⊢ (∀𝑥𝜑 → (𝑧 = 𝐴 ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑})))
5	4	bicomd 223	. . 3 ⊢ (∀𝑥𝜑 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ 𝑧 = 𝐴))
6	5	exbidv 1920	. 2 ⊢ (∀𝑥𝜑 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑧 𝑧 = 𝐴))
7		iseqsetv-clel 2823	. . 3 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
8	7	a1i 11	. 2 ⊢ (∀𝑥𝜑 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
9	2, 6, 8	3bitrd 305	1 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-clel 2819

This theorem is referenced by:  bj-issetw  36842