Theorem eqbrtrrd 3777

Description: Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Hypotheses

Ref	Expression
eqbrtrrd.1	⊢ (φ → A = B)
eqbrtrrd.2	⊢ (φ → A𝑅𝐶)

Assertion

Ref	Expression
eqbrtrrd	⊢ (φ → B𝑅𝐶)

Proof of Theorem eqbrtrrd

Step	Hyp	Ref	Expression
1		eqbrtrrd.1	. . 3 ⊢ (φ → A = B)
2	1	eqcomd 2042	. 2 ⊢ (φ → B = A)
3		eqbrtrrd.2	. 2 ⊢ (φ → A𝑅𝐶)
4	2, 3	eqbrtrd 3775	1 ⊢ (φ → B𝑅𝐶)

Syntax hints:   → wi 4   = wceq 1242   class class class wbr 3755

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756

This theorem is referenced by:  dftpos4  5819  prmuloclemcalc  6546  mullocprlem  6551  cauappcvgprlemladdfl  6627  caucvgprlemopl  6640  axarch  6773  lemulge11  7613  ltexp2a  8960  leexp2a  8961  nnlesq  9009