Theorem breqan12d 4045

Description: Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Hypotheses

Ref	Expression
breq1d.1	|- ( ph -> A = B )
breqan12i.2	|- ( ps -> C = D )

Assertion

Ref	Expression
breqan12d	|- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Proof of Theorem breqan12d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 |- ( ph -> A = B )
2		breqan12i.2	. 2 |- ( ps -> C = D )
3		breq12 4034	. 2 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   class class class wbr 4029

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  breqan12rd  4046  sosng  4732  isoresbr  5852  isoid  5853  isores3  5858  isoini2  5862  ofrfval  6139  oviec  6695  enqbreq2  7417  ltresr2  7900  axpre-ltadd  7946  leltadd  8466  xltneg  9902  lt2sq  10684  le2sq  10685  cnreim  11122  sqrtle  11180  sqrtlt  11181  absext  11207  reefiso  14912  logltb  15009