Theorem 3brtr3g 2636

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 2618	. 2 |- (ARB <-> CRD)
5	1, 4	sylib 198	1 |- (ph -> CRD)

Syntax hints:   -> wi 3   = wceq 953   class class class wbr 2609

This theorem is referenced by:  syl5eqbrr 2639  syl6breq 2644  ssenen 4484  prodgt0lem 5774  cvgratlem3ALT 7184  siilem1 8442

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610

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