Theorem breq123d 5125

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 5124	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4		breq123d.2	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
5	4	breqd 5122	. 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷))
6	3, 5	bitrd 282	1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   class class class wbr 5111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112

This theorem is used by:  sbcbr123  5167  fmptco  7130  xpsle  17660  invfuc  18061  yonedainv  18364  submomnd  20251  suborng  21034  opphllem3  29086  plngval  29115  lmif  29150  islmib  29152  iscgra  29176  isinag  29215  fmptcof2  33078  sgnsv  33549  inftmrel  33569  isinftm  33570  submarchi  33575  rlocval  33648  rprmval  33875  weiunval  37035  uncov  38314  iscvlat  40160  paddfval  40634  lhpset  40832  tendofset  41595  diaffval  41867  fnwe2val  43854  aomclem8  43866  afv2eq12d  48030