Theorem brneqtrd 39248

Description: Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
brneqtrd.1	|- ( ph -> -. A R B )
brneqtrd.2	|- ( ph -> B = C )

Assertion

Ref	Expression
brneqtrd	|- ( ph -> -. A R C )

Proof of Theorem brneqtrd

Step	Hyp	Ref	Expression
1		brneqtrd.1	. 2 |- ( ph -> -. A R B )
2		brneqtrd.2	. . 3 |- ( ph -> B = C )
3	2	breq2d 4665	. 2 |- ( ph -> ( A R B <-> A R C ) )
4	1, 3	mtbid 314	1 |- ( ph -> -. A R C )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by: (None)