Theorem hbop 1879

Description: Bound-variable hypothesis builder for ordered pairs.

Hypotheses

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

Assertion

Ref	Expression
hbop	⊢ (y ∈ ⟨A, B⟩ → ∀x y ∈ ⟨A, B⟩)

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

Proof of Theorem hbop

Step	Hyp	Ref	Expression
1		hbop.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	hbsn 1833	. . 3 ⊢ (y ∈ {A} → ∀x y ∈ {A})
3		hbop.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
4	1, 3	hbpr 1824	. . 3 ⊢ (y ∈ {A, B} → ∀x y ∈ {A, B})
5	2, 4	hbpr 1824	. 2 ⊢ (y ∈ {{A}, {A, B}} → ∀x y ∈ {{A}, {A, B}})
6		df-op 1815	. . 3 ⊢ ⟨A, B⟩ = {{A}, {A, B}}
7	6	eleq2i 1153	. 2 ⊢ (y ∈ ⟨A, B⟩ ↔ y ∈ {{A}, {A, B}})
8	7	bial 695	. 2 ⊢ (∀x y ∈ ⟨A, B⟩ ↔ ∀x y ∈ {{A}, {A, B}})
9	5, 7, 8	3imtr4 192	1 ⊢ (y ∈ ⟨A, B⟩ → ∀x y ∈ ⟨A, B⟩)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  {csn 1808  {cpr 1809  ⟨cop 1810

This theorem is referenced by:  moop2 1910  hbbr 2095  hbima 2609  hbopr 3017  xpmapenlem1 3391  seqlem1 4662

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

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