Theorem iuneq2dv 3986

Description: Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)

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

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem iuneq2dv

Step	Hyp	Ref	Expression
1		iuneq2dv.1	. . 3 |- ( ( ph /\ x e. A ) -> B = C )
2	1	ralrimiva 2603	. 2 |- ( ph -> A. x e. A B = C )
3		iuneq2 3981	. 2 |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
4	2, 3	syl 14	1 |- ( ph -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   U_ciun 3965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-iun 3967

This theorem is referenced by:  iuneq12d  3989  iuneq2d  3990  oav2  6609  omv2  6611  ennnfonelemrn  12990  tgidm  14748