Theorem breq1d 4650

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

Hypothesis

Ref	Expression
breq1d.1	|- (ph -> A = B)

Assertion

Ref	Expression
breq1d	|- (ph -> (ARC <-> BRC))

Proof of Theorem breq1d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 |- (ph -> A = B)
2		breq1 4643	. 2 |- (A = B -> (ARC <-> BRC))
3	1, 2	syl 15	1 |- (ph -> (ARC <-> BRC))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641

This theorem is referenced by:  eqbrtrd  4660  syl6eqbr  4677  sbcbr2g  4689  br1stg  4731  isorel  5490  isocnv  5492  isotr  5496  caovord  5630  qrpprod  5837  xpsneng  6051  enpw1  6063  enmap2  6069  enpw  6088  mucnc  6132  ce2le  6234  nchoicelem11  6300