Theorem rabeqf 2853

Description: Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)

Hypotheses

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

Assertion

Ref	Expression
rabeqf	⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 ⊢ ℲxA
2		rabeqf.2	. . . 4 ⊢ ℲxB
3	1, 2	nfeq 2497	. . 3 ⊢ Ⅎx A = B
4		eleq2 2414	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 685	. . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
6	3, 5	abbid 2467	. 2 ⊢ (A = B → {x ∣ (x ∈ A ∧ φ)} = {x ∣ (x ∈ B ∧ φ)})
7		df-rab 2624	. 2 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
8		df-rab 2624	. 2 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)}
9	6, 7, 8	3eqtr4g 2410	1 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ})

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  {crab 2619

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624

This theorem is used by:  rabeq  2854