Theorem ceqsralv 2802

Description: Restricted quantifier version of ceqsalv 2801. (Contributed by NM, 21-Jun-2013.)

Hypothesis

Ref	Expression
ceqsralv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ceqsralv	⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsralv

Step	Hyp	Ref	Expression
1		nfv 1550	. 2 ⊢ Ⅎ𝑥𝜓
2		ceqsralv.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	ax-gen 1471	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		ceqsralt 2798	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
5	1, 3, 4	mp3an12 1339	1 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1370   = wceq 1372  Ⅎwnf 1482   ∈ wcel 2175  ∀wral 2483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-v 2773

This theorem is referenced by:  eqreu  2964  sqrt2irr  12426