Theorem uneq1d 3135

Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Hypothesis

Ref	Expression
uneq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
uneq1d	|- ( ph -> ( A u. C ) = ( B u. C ) )

Proof of Theorem uneq1d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 |- ( ph -> A = B )
2		uneq1 3129	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   = wceq 1285   u. cun 2980

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986

This theorem is referenced by:  ifeq1  3371  preq1  3487  tpeq1  3496  tpeq2  3497  resasplitss  5120  fmptpr  5407  rdgisucinc  6054  oasuc  6128  omsuc  6136  fisseneq  6474  fzpred  9215  fseq1p1m1  9239  nn0split  9276  fzo0sn0fzo1  9359  fzosplitprm1  9372  zsupcllemstep  10548