Theorem iuneq2dv 2650

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:   φ,x

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 ⊢ ((φ ⋀ x ∈ A) → B = C)
2	1	r19.21aiva 1760	. 2 ⊢ (φ → ∀x ∈ A B = C)
3		iuneq2 2646	. 2 ⊢ (∀x ∈ A B = C → ∪x ∈ A B = ∪x ∈ A C)
4	2, 3	syl 10	1 ⊢ (φ → ∪x ∈ A B = ∪x ∈ A C)

Syntax hints:   → wi 3   ⋀ wa 221   = wceq 992   ∈ wcel 994  ∀wral 1691  ∪ciun 2633

This theorem is referenced by:  fparlem3 4201  fparlem4 4202  oalim 4303  omlim 4304  oelim 4305  oelim2 4358  cncnplem4 7987  imfstnrelc 10810  compsub 11488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ral 1695  df-rex 1696  df-v 1858  df-in 2103  df-ss 2105  df-iun 2635

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