Theorem ralsns 3763

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Assertion

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

Distinct variable group:   x,A

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

Proof of Theorem ralsns

Step	Hyp	Ref	Expression
1		sbc6g 3071	. 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∀x(x = A → φ)))
2		df-ral 2619	. . 3 ⊢ (∀x ∈ {A}φ ↔ ∀x(x ∈ {A} → φ))
3		elsn 3748	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
4	3	imbi1i 315	. . . 4 ⊢ ((x ∈ {A} → φ) ↔ (x = A → φ))
5	4	albii 1566	. . 3 ⊢ (∀x(x ∈ {A} → φ) ↔ ∀x(x = A → φ))
6	2, 5	bitri 240	. 2 ⊢ (∀x ∈ {A}φ ↔ ∀x(x = A → φ))
7	1, 6	syl6rbbr 255	1 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ [̣A / x]̣φ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614  [̣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-ral 2619  df-v 2861  df-sbc 3047  df-sn 3741

This theorem is referenced by:  ralsng  3765  sbcsng  3783