Theorem 3brtr3g 4077

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 4053	. 2 ⊢ (𝐴𝑅𝐵 ↔ 𝐶𝑅𝐷)
5	1, 4	sylib 122	1 ⊢ (𝜑 → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   = wceq 1373   class class class wbr 4044

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by:  eqbrtrrid  4080  breqtrdi  4085  ssenen  6948  endjusym  7198  djuen  7323  ege2le3  11982