Theorem rabeqf 1345

Description: Equality theorem for restricted class abstractions, with bound variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
rabeqf.1	⊢ (y ∈ A → ∀x y ∈ A)
rabeqf.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

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

Distinct variable group(s):   y,A   y,B   x,y

Proof of Theorem rabeqf

Step	Hyp	Ref	Expression
1		rabeqf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		rabeqf.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1171	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1150	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 469	. . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
6	3, 5	biabd 1182	. 2 ⊢ (A = B → {x∣(x ∈ A ∧ φ)} = {x∣(x ∈ B ∧ φ)})
7		df-rab 1208	. 2 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
8		df-rab 1208	. 2 ⊢ {x ∈ B∣φ} = {x∣(x ∈ B ∧ φ)}
9	6, 7, 8	3eqtr4g 1147	1 ⊢ (A = B → {x ∈ A∣φ} = {x ∈ B∣φ})

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  {crab 1204

This theorem is referenced by:  rabeq 1346  hta 3619

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208

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