Theorem iuneq2dv 2586

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 1717	. 2 |- (ph -> A.x e. A B = C)
3		iuneq2 2582	. 2 |- (A.x e. A B = C -> U_x e. A B = U_x e. A C)
4	2, 3	syl 10	1 |- (ph -> U_x e. A B = U_x e. A C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  U_ciun 2570

This theorem is referenced by:  oalim 4173  omlim 4174  oelim 4175  oelim2 4228  cncnplem4 7774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-rex 1653  df-v 1815  df-in 2054  df-ss 2056  df-iun 2572

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