Theorem iuneq2dv 2010

Description: Equality deduction for indexed union.

Hypothesis

Ref	Expression
iuneq2dv.1	⊢ (φ → (x ∈ A → B = C))

Assertion

Ref	Expression
iuneq2dv	⊢ (φ → ∪x ∈ A B = ∪x ∈ A C)

Distinct variable group(s):   φ,x

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 ⊢ (φ → (x ∈ A → B = C))
2	1	r19.21aiv 1259	. 2 ⊢ (φ → ∀x ∈ A B = C)
3		iuneq2 2006	. 2 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)
4	2, 3	syl 12	1 ⊢ (φ → ∪x ∈ A B = ∪x ∈ A C)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∪ciun 1994

This theorem is referenced by:  oalim 3135  omlim 3136  oelim 3137

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-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-iun 1996

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