Theorem opeq1i 2481

Description: Equality inference for ordered pairs.

Hypothesis

Ref	Expression
opeq1i.1	|- A = B

Assertion

Ref	Expression
opeq1i	|- <.A, C>. = <.B, C>.

Proof of Theorem opeq1i

Step	Hyp	Ref	Expression
1		opeq1i.1	. 2 |- A = B
2		opeq1 2478	. 2 |- (A = B -> <.A, C>. = <.B, C>.)
3	1, 2	ax-mp 7	1 |- <.A, C>. = <.B, C>.

Syntax hints:   = wceq 953  <.cop 2401

This theorem is referenced by:  xpmapenlem2 4477  ltexpq 5052  halfpq 5054  axi2m1 5257  isumnn0nn 7142  geolim1i 7173  efseq0ex 7253  ef1tllem 7323  efm1lim 7351  indistps 7595  distps 7596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406

Copyright terms: Public domain