Theorem opeq1i 3161

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 3158	. 2 |- (A = B -> <.A, C>. = <.B, C>.)
3	1, 2	ax-mp 7	1 |- <.A, C>. = <.B, C>.

Syntax hints:   = wceq 1298  <.cop 3046

This theorem is referenced by:  xpmapenlem2 5591  ltexpq 6232  halfpq 6234  axi2m1 6438  isumnn0nn 8468  geolim1i 8500  efseq0ex 8573  ef1tllem 8643  efm1limi 8676  indistps 8923  distps 8924  fsumltisumi 15823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053

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