Theorem ralsn 3768

Description: Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Hypotheses

Ref	Expression
ralsn.1	⊢ A ∈ V
ralsn.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ralsn	⊢ (∀x ∈ {A}φ ↔ ψ)

Distinct variable groups:   x,A   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem ralsn

Step	Hyp	Ref	Expression
1		ralsn.1	. 2 ⊢ A ∈ V
2		ralsn.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	ralsng 3766	. 2 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ ψ))
4	1, 3	ax-mp 5	1 ⊢ (∀x ∈ {A}φ ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860  {csn 3738

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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-sbc 3048  df-sn 3742

This theorem is referenced by: (None)