Theorem breq123d 4047

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 4046	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4		breq123d.2	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
5	4	breqd 4044	. 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷))
6	3, 5	bitrd 188	1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   class class class wbr 4033

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  sbcbrg  4087  fmptco  5728