Theorem ralsng 3766

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
ralsng.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ralsng	⊢ (A ∈ V → (∀x ∈ {A}φ ↔ ψ))

Distinct variable groups:   x,A   ψ,x

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

Proof of Theorem ralsng

Step	Hyp	Ref	Expression
1		ralsns 3764	. 2 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ [̣A / x]̣φ))
2		ralsng.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	sbcieg 3079	. 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ψ))
4	1, 3	bitrd 244	1 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2615  [̣wsbc 3047  {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:  ralsn  3768  ralprg  3776  raltpg  3778  ralunsn  3880  iinxsng  4043