Theorem ralsngOLD 4686

Description: Obsolete version of ralsng 4682 as of 30-Sep-2024. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 7-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

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

Proof of Theorem ralsngOLD

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜓
2		ralsngOLD.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	ralsngf 4680	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ∀wral 3058  {csn 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3475  df-sbc 3779  df-sn 4633

This theorem is referenced by: (None)