Theorem iuneq2d 4016

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 4012	1 |- ( ph -> U_ x e. A B = U_ x e. A C )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   U_ciun 3991

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-in 3217  df-ss 3224  df-iun 3993

This theorem is referenced by:  rdgeq1  6602  imasex  13518