Theorem breq123d 4097

Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
breq123d.2	⊢ (𝜑 → 𝑅 = 𝑆)
breq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
breq123d	⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Proof of Theorem breq123d

Step	Hyp	Ref	Expression
1		breq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		breq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	breq12d 4096	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4		breq123d.2	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
5	4	breqd 4094	. 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷))
6	3, 5	bitrd 188	1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   class class class wbr 4083

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by:  sbcbrg  4138  fmptco  5801