Theorem 3brtr4g 4122

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

Hypotheses

Ref	Expression
3brtr4g.1	⊢ (𝜑 → 𝐴𝑅𝐵)
3brtr4g.2	⊢ 𝐶 = 𝐴
3brtr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3brtr4g	⊢ (𝜑 → 𝐶𝑅𝐷)

Proof of Theorem 3brtr4g

Step	Hyp	Ref	Expression
1		3brtr4g.1	. 2 ⊢ (𝜑 → 𝐴𝑅𝐵)
2		3brtr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3brtr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	breq12i 4097	. 2 ⊢ (𝐶𝑅𝐷 ↔ 𝐴𝑅𝐵)
5	1, 4	sylibr 134	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1397   class class class wbr 4088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089

This theorem is referenced by:  eqbrtrid  4123  enpr2d  6996  crth  12795  4sqlem6  12955  gausslemma2dlem0f  15782  gausslemma2dlem0g  15783  wlkcompim  16202  upgrwlkcompim  16212  trilpolemgt1  16643