Theorem iuneq1d 4030

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 4020	. 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 1402   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  iuneq12d  4031