Theorem rexsns 4671

Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)

Assertion

Ref	Expression
rexsns	⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsns

Step	Hyp	Ref	Expression
1		velsn 4642	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	anbi1i 624	. . 3 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜑))
3	2	exbii 1848	. 2 ⊢ (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
4		df-rex 3071	. 2 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
5		sbc5 3816	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
6	3, 4, 5	3bitr4i 303	1 ⊢ (∃𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070  [wsbc 3788  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-sbc 3789  df-sn 4627

This theorem is referenced by:  rexsngf  4672  r19.12sn  4720  poimirlem25  37652