Theorem 3brtr3g 5141

Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 16-Jan-1997.)

Hypotheses

Ref	Expression
3brtr3g.1	⊢ (𝜑 → 𝐴𝑅𝐵)
3brtr3g.2	⊢ 𝐴 = 𝐶
3brtr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3brtr3g	⊢ (𝜑 → 𝐶𝑅𝐷)

Proof of Theorem 3brtr3g

Step	Hyp	Ref	Expression
1		3brtr3g.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		3brtr3g.2	. . 3 ⊢ 𝐴 = 𝐶
3		3brtr3g.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	breq12i 5115	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)
5	1, 4	sylib 221	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5106

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107

This theorem is used by:  eqbrtrrid  5144  breqtrdi  5149  ssenen  9195  adderpq  11065  mulerpq  11066  ltaddnq  11083  ege2le3  16282  omndaddr  20370  ogrpaddltrd  20381  ovolfiniun  25849  dvfsumlem3  26375  basellem9  27443  pnt2  27967  pnt  27968  siilem1  31480  sn-0ne2  43648