Theorem iuneq1d 3889

Description: Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Hypothesis

Ref	Expression
iuneq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
iuneq1d	|- ( ph -> U_ x e. A C = U_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem iuneq1d

Step	Hyp	Ref	Expression
1		iuneq1d.1	. 2 |- ( ph -> A = B )
2		iuneq1 3879	. 2 |- ( A = B -> U_ x e. A C = U_ x e. B C )
3	1, 2	syl 14	1 |- ( ph -> U_ x e. A C = U_ x e. B C )

Syntax hints:   -> wi 4   = wceq 1343   U_ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-iun 3868

This theorem is referenced by:  iuneq12d  3890