Theorem hbpr 1824

Description: Bound-variable hypothesis builder for unordered pairs.

Hypotheses

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

Assertion

Ref	Expression
hbpr	⊢ (y ∈ {A, B} → ∀x y ∈ {A, B})

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

Proof of Theorem hbpr

Step	Hyp	Ref	Expression
1		hbpr.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	hbeleq 1173	. . 3 ⊢ (y = A → ∀x y = A)
3		hppr.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
4	3	hbeleq 1173	. . 3 ⊢ (y = B → ∀x y = B)
5	2, 4	hbor 703	. 2 ⊢ ((y = A ∨ y = B) → ∀x(y = A ∨ y = B))
6		visset 1350	. . 3 ⊢ y ∈ V
7	6	elpr 1823	. 2 ⊢ (y ∈ {A, B} ↔ (y = A ∨ y = B))
8	7	bial 695	. 2 ⊢ (∀x y ∈ {A, B} ↔ ∀x(y = A ∨ y = B))
9	5, 7, 8	3imtr4 192	1 ⊢ (y ∈ {A, B} → ∀x y ∈ {A, B})

Syntax hints:   → wi 2   ∨ wo 195  ∀wal 672   = wceq 1091   ∈ wcel 1092  {cpr 1809

This theorem is referenced by:  hbsn 1833  hbop 1879

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

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