Theorem 3brtr3g 2646

Description: Substitution of equality into both sides of a binary relation.

Hypotheses

Ref	Expression
3brtr3g.1	|- (ph -> ARB)
3brtr3g.2	|- A = C
3brtr3g.3	|- B = D

Assertion

Ref	Expression
3brtr3g	|- (ph -> CRD)

Proof of Theorem 3brtr3g

Step	Hyp	Ref	Expression
1		3brtr3g.1	. 2 |- (ph -> ARB)
2		3brtr3g.2	. . 3 |- A = C
3		3brtr3g.3	. . 3 |- B = D
4	2, 3	breq12i 2628	. 2 |- (ARB <-> CRD)
5	1, 4	sylib 198	1 |- (ph -> CRD)

Syntax hints:   -> wi 3   = wceq 956   class class class wbr 2619

This theorem is referenced by:  syl5eqbrr 2649  syl6breq 2654  ssenen 4504  prodgt0lem 5818  cvgratlem3ALT 7249  siilem1 8511

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  df-br 2620

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