Theorem raleqf 2804

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleq1f.1	⊢ ℲxA
raleq1f.2	⊢ ℲxB

Assertion

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

Proof of Theorem raleqf

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ ℲxA
2		raleq1f.2	. . . 4 ⊢ ℲxB
3	1, 2	nfeq 2497	. . 3 ⊢ Ⅎx A = B
4		eleq2 2414	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	imbi1d 308	. . 3 ⊢ (A = B → ((x ∈ A → φ) ↔ (x ∈ B → φ)))
6	3, 5	albid 1772	. 2 ⊢ (A = B → (∀x(x ∈ A → φ) ↔ ∀x(x ∈ B → φ)))
7		df-ral 2620	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
8		df-ral 2620	. 2 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
9	6, 7, 8	3bitr4g 279	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is used by:  raleq  2808