Theorem hbun 1614

Description: Bound-variable hypothesis builder for the union of classes.

Hypotheses

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

Assertion

Ref	Expression
hbun	⊢ (y ∈ (A ∪ B) → ∀x y ∈ (A ∪ B))

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

Proof of Theorem hbun

Step	Hyp	Ref	Expression
1		hbun.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbun.2	. . 3 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbor 703	. 2 ⊢ ((y ∈ A ∨ y ∈ B) → ∀x(y ∈ A ∨ y ∈ B))
4		elun 1601	. 2 ⊢ (y ∈ (A ∪ B) ↔ (y ∈ A ∨ y ∈ B))
5	4	bial 695	. 2 ⊢ (∀x y ∈ (A ∪ B) ↔ ∀x(y ∈ A ∨ y ∈ B))
6	3, 4, 5	3imtr4 192	1 ⊢ (y ∈ (A ∪ B) → ∀x y ∈ (A ∪ B))

Syntax hints:   → wi 2   ∨ wo 195  ∀wal 672   ∈ wcel 1092   ∪ cun 1485

This theorem is referenced by:  trcl 3489

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

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