Theorem 3brtr3g 5175

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 5151	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)
5	1, 4	sylib 218	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1539   class class class wbr 5142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143

This theorem is referenced by:  eqbrtrrid  5178  breqtrdi  5183  ssenen  9192  adderpq  10997  mulerpq  10998  ltaddnq  11015  ege2le3  16127  ovolfiniun  25537  dvfsumlem3  26070  basellem9  27133  pnt2  27658  pnt  27659  siilem1  30871  omndaddr  33085  ogrpaddltrd  33097  sn-0ne2  42441