Theorem eqbrriv 3249

Description: Inference from extensionality principle for relations.

Hypotheses

Ref	Expression
eqbrriv.1	|- Rel A
eqbrriv.2	|- Rel B
eqbrriv.3	|- (xAy <-> xBy)

Assertion

Ref	Expression
eqbrriv	|- A = B

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem eqbrriv

Step	Hyp	Ref	Expression
1		eqbrriv.1	. 2 |- Rel A
2		eqbrriv.2	. 2 |- Rel B
3		eqbrriv.3	. . 3 |- (xAy <-> xBy)
4		df-br 2617	. . 3 |- (xAy <-> <.x, y>. e. A)
5		df-br 2617	. . 3 |- (xBy <-> <.x, y>. e. B)
6	3, 4, 5	3bitr3 181	. 2 |- (<.x, y>. e. A <-> <.x, y>. e. B)
7	1, 2, 6	eqrelriv 3248	1 |- A = B

Syntax hints:   <-> wb 146   = wceq 955   e. wcel 957  <.cop 2409   class class class wbr 2616  Rel wrel 3172

This theorem is referenced by:  resco 3497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-xp 3181  df-rel 3182

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