Theorem ceqsralt 2883

Description: Restricted quantifier version of ceqsalt 2882. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Assertion

Ref	Expression
ceqsralt	⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ ψ))

Distinct variable groups:   x,A   x,B

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

Proof of Theorem ceqsralt

Step	Hyp	Ref	Expression
1		df-ral 2620	. . . 4 ⊢ (∀x ∈ B (x = A → φ) ↔ ∀x(x ∈ B → (x = A → φ)))
2		eleq1 2413	. . . . . . . . 9 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	2	pm5.32ri 619	. . . . . . . 8 ⊢ ((x ∈ B ∧ x = A) ↔ (A ∈ B ∧ x = A))
4	3	imbi1i 315	. . . . . . 7 ⊢ (((x ∈ B ∧ x = A) → φ) ↔ ((A ∈ B ∧ x = A) → φ))
5		impexp 433	. . . . . . 7 ⊢ (((x ∈ B ∧ x = A) → φ) ↔ (x ∈ B → (x = A → φ)))
6		impexp 433	. . . . . . 7 ⊢ (((A ∈ B ∧ x = A) → φ) ↔ (A ∈ B → (x = A → φ)))
7	4, 5, 6	3bitr3i 266	. . . . . 6 ⊢ ((x ∈ B → (x = A → φ)) ↔ (A ∈ B → (x = A → φ)))
8	7	albii 1566	. . . . 5 ⊢ (∀x(x ∈ B → (x = A → φ)) ↔ ∀x(A ∈ B → (x = A → φ)))
9	8	a1i 10	. . . 4 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x(x ∈ B → (x = A → φ)) ↔ ∀x(A ∈ B → (x = A → φ))))
10	1, 9	syl5bb 248	. . 3 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ ∀x(A ∈ B → (x = A → φ))))
11		19.21v 1890	. . 3 ⊢ (∀x(A ∈ B → (x = A → φ)) ↔ (A ∈ B → ∀x(x = A → φ)))
12	10, 11	syl6bb 252	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ (A ∈ B → ∀x(x = A → φ))))
13		biimt 325	. . 3 ⊢ (A ∈ B → (∀x(x = A → φ) ↔ (A ∈ B → ∀x(x = A → φ))))
14	13	3ad2ant3 978	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x(x = A → φ) ↔ (A ∈ B → ∀x(x = A → φ))))
15		ceqsalt 2882	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x(x = A → φ) ↔ ψ))
16	12, 14, 15	3bitr2d 272	1 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ)) ∧ A ∈ B) → (∀x ∈ B (x = A → φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2615

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 2620  df-v 2862

This theorem is referenced by:  ceqsralv  2887