Theorem rexsns 3764

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

Assertion

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

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem rexsns

Step	Hyp	Ref	Expression
1		sbc5 3070	. . 3 ⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ))
2	1	a1i 10	. 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ)))
3		df-rex 2620	. . 3 ⊢ (∃x ∈ {A}φ ↔ ∃x(x ∈ {A} ∧ φ))
4		elsn 3748	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
5	4	anbi1i 676	. . . 4 ⊢ ((x ∈ {A} ∧ φ) ↔ (x = A ∧ φ))
6	5	exbii 1582	. . 3 ⊢ (∃x(x ∈ {A} ∧ φ) ↔ ∃x(x = A ∧ φ))
7	3, 6	bitri 240	. 2 ⊢ (∃x ∈ {A}φ ↔ ∃x(x = A ∧ φ))
8	2, 7	syl6rbbr 255	1 ⊢ (A ∈ V → (∃x ∈ {A}φ ↔ [̣A / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  [̣wsbc 3046  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741

This theorem is referenced by:  rexsng  3766  r19.12sn  3789