Theorem hbiu1 2012

Description: Bound-variable hypothesis builder for indexed union.

Assertion

Ref	Expression
hbiu1	⊢ (y ∈ ∪x ∈ A B → ∀x y ∈ ∪x ∈ A B)

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

Proof of Theorem hbiu1

Step	Hyp	Ref	Expression
1		hbre1 1239	. . 3 ⊢ (∃x ∈ A z ∈ B → ∀x∃x ∈ A z ∈ B)
2	1	hbab 1096	. 2 ⊢ (y ∈ {z∣∃x ∈ A z ∈ B} → ∀x y ∈ {z∣∃x ∈ A z ∈ B})
3		df-iun 1996	. . 3 ⊢ ∪x ∈ A B = {z∣∃x ∈ A z ∈ B}
4	3	eleq2i 1153	. 2 ⊢ (y ∈ ∪x ∈ A B ↔ y ∈ {z∣∃x ∈ A z ∈ B})
5	4	bial 695	. 2 ⊢ (∀x y ∈ ∪x ∈ A B ↔ ∀x y ∈ {z∣∃x ∈ A z ∈ B})
6	2, 4, 5	3imtr4 192	1 ⊢ (y ∈ ∪x ∈ A B → ∀x y ∈ ∪x ∈ A B)

Syntax hints:   → wi 2  ∀wal 672  {cab 1090   ∈ wcel 1092  ∃wrex 1202  ∪ciun 1994

This theorem is referenced by:  ssiun2s 2020  r1val1 3502  rankuni 3533  ranklon 3540  cplem2 3546

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-rex 1206  df-iun 1996

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