Theorem iuneq2d 3874

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

Hypothesis

Ref	Expression
iuneq2d.2	|- ( ph -> B = C )

Assertion

Ref	Expression
iuneq2d	|- ( ph -> U_ x e. A B = U_ x e. A C )

Distinct variable groups:   ph, x   x, A

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

Proof of Theorem iuneq2d

Step	Hyp	Ref	Expression
1		iuneq2d.2	. . 3 |- ( ph -> B = C )
2	1	adantr 274	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	2	iuneq2dv 3870	1 |- ( ph -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   = wceq 1335   e. wcel 2128   U_ciun 3849

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-iun 3851

This theorem is referenced by:  rdgeq1  6312