Theorem hbrab1 1310

Description: The abstraction variable in a restricted class abstraction isn't free.

Assertion

Ref	Expression
hbrab1	⊢ (y ∈ {x ∈ A∣φ} → ∀x y ∈ {x ∈ A∣φ})

Distinct variable group(s):   x,y

Proof of Theorem hbrab1

Step	Hyp	Ref	Expression
1		hbab1 1095	. 2 ⊢ (y ∈ {x∣(x ∈ A ∧ φ)} → ∀x y ∈ {x∣(x ∈ A ∧ φ)})
2		df-rab 1208	. . 3 ⊢ {x ∈ A∣φ} = {x∣(x ∈ A ∧ φ)}
3	2	eleq2i 1153	. 2 ⊢ (y ∈ {x ∈ A∣φ} ↔ y ∈ {x∣(x ∈ A ∧ φ)})
4	3	bial 695	. 2 ⊢ (∀x y ∈ {x ∈ A∣φ} ↔ ∀x y ∈ {x∣(x ∈ A ∧ φ)})
5	1, 3, 4	3imtr4 192	1 ⊢ (y ∈ {x ∈ A∣φ} → ∀x y ∈ {x ∈ A∣φ})

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

This theorem is referenced by:  reuuni2 1956  reuuni4 1959  onminsb 2264  oawordeulem 3156  tz9.12lem3 3505  rankid 3516  ondomcard 3663  cardmin 3666  alephordlem1 3677  cardaleph 3690  nnwos 4610

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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