Theorem iuneq2dv 3279

Description: Equality deduction for indexed union.

Hypothesis

Ref	Expression
iuneq2dv.1	|- ((ph /\ x e. A) -> B = C)

Assertion

Ref	Expression
iuneq2dv	|- (ph -> U_x e. A B = U_x e. A C)

Distinct variable group:   ph,x

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 |- ((ph /\ x e. A) -> B = C)
2	1	r19.21aiva 2176	. 2 |- (ph -> A.x e. A B = C)
3		iuneq2 3273	. 2 |- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
4	2, 3	syl 12	1 |- (ph -> U_x e. A B = U_x e. A C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  U_ciun 3255

This theorem is referenced by:  fparlem3 5083  fparlem4 5084  oalim 5212  omlim 5213  oelim 5214  oelim2 5270  cncnplem4 9054  trcleq1 13926  trcleq2 13927  imfstnrelc 14396  trclval 15271  compsub 15431

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-iun 3257

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