Theorem breq123d 5112

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 5111	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4		breq123d.2	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
5	4	breqd 5109	. 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷))
6	3, 5	bitrd 279	1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   class class class wbr 5098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099

This theorem is referenced by:  sbcbr123  5152  fmptco  7074  xpsle  17500  invfuc  17901  yonedainv  18204  submomnd  20061  suborng  20809  opphllem3  28821  lmif  28857  islmib  28859  iscgra  28881  isinag  28910  fmptcof2  32735  sgnsv  33242  inftmrel  33262  isinftm  33263  submarchi  33268  rlocval  33341  rprmval  33597  weiunval  36656  uncov  37802  iscvlat  39593  paddfval  40067  lhpset  40265  tendofset  41028  diaffval  41300  fnwe2val  43301  aomclem8  43313  afv2eq12d  47471