Theorem ceqsralv 2886

Description: Restricted quantifier version of ceqsalv 2885. (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 1619	. 2 ⊢ Ⅎxψ
2		ceqsralv.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	ax-gen 1546	. 2 ⊢ ∀x(x = A → (φ ↔ ψ))
4		ceqsralt 2882	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ ψ))
5	1, 3, 4	mp3an12 1267	1 ⊢ (A ∈ B → (∀x ∈ B (x = A → φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-v 2861

This theorem is referenced by:  eqreu  3028