Theorem iuneq2d 3937

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 276	. 2 |- ( ( ph /\ x e. A ) -> B = C )
3	2	iuneq2dv 3933	1 |- ( ph -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   U_ciun 3912

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-iun 3914

This theorem is referenced by:  rdgeq1  6415  imasex  12878