Theorem eqbrriv 3252

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 2620	. . 3 |- (xAy <-> <.x, y>. e. A)
5		df-br 2620	. . 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 3251	1 |- A = B

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  <.cop 2411   class class class wbr 2619  Rel wrel 3175

This theorem is referenced by:  resco 3500

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-11 967  ax-12 968  ax-13 969  ax-14 970  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  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185

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