Theorem iuneq2i 3878

Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Hypothesis

Ref	Expression
iuneq2i.1	|- ( x e. A -> B = C )

Assertion

Ref	Expression
iuneq2i	|- U_ x e. A B = U_ x e. A C

Proof of Theorem iuneq2i

Step	Hyp	Ref	Expression
1		iuneq2 3876	. 2 |- ( A. x e. A B = C -> U_ x e. A B = U_ x e. A C )
2		iuneq2i.1	. 2 |- ( x e. A -> B = C )
3	1, 2	mprg 2521	1 |- U_ x e. A B = U_ x e. A C

Syntax hints:   -> wi 4   = wceq 1342   e. wcel 2135   U_ciun 3860

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-in 3117  df-ss 3124  df-iun 3862

This theorem is referenced by:  dfiunv2  3896  iunrab  3907  iunid  3915  iunin1  3924  2iunin  3926  resiun1  4897  resiun2  4898  dfimafn2  5530  dfmpt  5656  rdgival  6341  uniqs  6550  txbasval  12808