Theorem 3brtr3g 5143

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 5119	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)
5	1, 4	sylib 218	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1540   class class class wbr 5110

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111

This theorem is referenced by:  eqbrtrrid  5146  breqtrdi  5151  ssenen  9121  adderpq  10916  mulerpq  10917  ltaddnq  10934  ege2le3  16063  ovolfiniun  25409  dvfsumlem3  25942  basellem9  27006  pnt2  27531  pnt  27532  siilem1  30787  omndaddr  33028  ogrpaddltrd  33040  sn-0ne2  42401