Theorem opeq1d 2493

Description: Equality deduction for ordered pairs.

Hypothesis

Ref	Expression
opeq1d.1	|- (ph -> A = B)

Assertion

Ref	Expression
opeq1d	|- (ph -> <.A, C>. = <.B, C>.)

Proof of Theorem opeq1d

Step	Hyp	Ref	Expression
1		opeq1d.1	. 2 |- (ph -> A = B)
2		opeq1 2487	. 2 |- (A = B -> <.A, C>. = <.B, C>.)
3	1, 2	syl 10	1 |- (ph -> <.A, C>. = <.B, C>.)

Syntax hints:   -> wi 3   = wceq 956  <.cop 2411

This theorem is referenced by:  opeq12d 2495  hbopd 2497  moop2 2801  cbvoprab12 3998  eloprabi 4118  iserzex 7146  nvop 8305  phop 8477  isded 10669  dedi 10670  iscat 10687  cati 10688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416

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