Theorem 3brtr3g 5125

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 5101	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)
5	1, 4	sylib 218	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093

This theorem is referenced by:  eqbrtrrid  5128  breqtrdi  5133  ssenen  9068  adderpq  10850  mulerpq  10851  ltaddnq  10868  ege2le3  15997  omndaddr  20008  ogrpaddltrd  20019  ovolfiniun  25400  dvfsumlem3  25933  basellem9  26997  pnt2  27522  pnt  27523  siilem1  30795  sn-0ne2  42389