Theorem breqan12d 3354

Description: Equality deduction for a binary relation.

Hypotheses

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

Assertion

Ref	Expression
breqan12d	|- ((ph /\ ps) -> (ARC <-> BRD))

Proof of Theorem breqan12d

Step	Hyp	Ref	Expression
1		breq12 3343	. 2 |- ((A = B /\ C = D) -> (ARC <-> BRD))
2		breq1d.1	. 2 |- (ph -> A = B)
3		breqan12i.2	. 2 |- (ps -> C = D)
4	1, 2, 3	syl2an 503	1 |- ((ph /\ ps) -> (ARC <-> BRD))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   class class class wbr 3338

This theorem is referenced by:  breqan12rd 3355  isoid 4872  isotr 4874  isotrALT 4875  oprec 5377  pre-axltadd 6442  leltadd 6830  lemul1a 7019  lemul1aOLD 7020  expwordi 7848  lt2sq 7875  le2sq 7876  sqrlei 7957  sqrlti 7958  minveclem26 9915  minveclem27 9916  logltb 10130  projlemHIL 10851  mddmd 11873  pi1f 16093  pi1val 16094

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339

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