Theorem bj-issetwt 32053

Description: Closed form of bj-issetw 32054. (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		df-clel 2606	. . 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		bj-vexwvt 32050	. . . . 5 ⊢ (∀𝑥𝜑 → 𝑧 ∈ {𝑥 ∣ 𝜑})
4	3	biantrud 527	. . . 4 ⊢ (∀𝑥𝜑 → (𝑧 = 𝐴 ↔ (𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑})))
5	4	bicomd 212	. . 3 ⊢ (∀𝑥𝜑 → ((𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ 𝑧 = 𝐴))
6	5	exbidv 1837	. 2 ⊢ (∀𝑥𝜑 → (∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ {𝑥 ∣ 𝜑}) ↔ ∃𝑧 𝑧 = 𝐴))
7		bj-denotes 32052	. . 3 ⊢ (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
8	7	a1i 11	. 2 ⊢ (∀𝑥𝜑 → (∃𝑧 𝑧 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
9	2, 6, 8	3bitrd 293	1 ⊢ (∀𝑥𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑦 𝑦 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-clel 2606

This theorem is referenced by:  bj-issetw  32054