Theorem hbbr 2095

Description: Bound-variable hypothesis builder for binary relation.

Hypotheses

Ref	Expression
hbbr.1	⊢ (y ∈ A → ∀x y ∈ A)
hbbr.2	⊢ (y ∈ R → ∀x y ∈ R)
hbbr.3	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbbr	⊢ (ARB → ∀x ARB)

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

Proof of Theorem hbbr

Step	Hyp	Ref	Expression
1		hbbr.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbbr.3	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbop 1879	. . 3 ⊢ (y ∈ ⟨A, B⟩ → ∀x y ∈ ⟨A, B⟩)
4		hbbr.2	. . 3 ⊢ (y ∈ R → ∀x y ∈ R)
5	3, 4	hbel 1172	. 2 ⊢ (⟨A, B⟩ ∈ R → ∀x⟨A, B⟩ ∈ R)
6		df-br 2063	. 2 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
7	6	bial 695	. 2 ⊢ (∀x ARB ↔ ∀x⟨A, B⟩ ∈ R)
8	5, 6, 7	3imtr4 192	1 ⊢ (ARB → ∀x ARB)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054

This theorem is referenced by:  hbco 2508  hbcnv 2516  dffunmof 2678  hbiso 2930  ondomcard 3663  cardmin 3666  alephordlem1 3677

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-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063

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