Theorem breq2d 4651

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

Hypothesis

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

Assertion

Ref	Expression
breq2d	|- (ph -> (CRA <-> CRB))

Proof of Theorem breq2d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 |- (ph -> A = B)
2		breq2 4643	. 2 |- (A = B -> (CRA <-> CRB))
3	1, 2	syl 15	1 |- (ph -> (CRA <-> CRB))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   class class class wbr 4639

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640

This theorem is referenced by:  breqtrd  4663  sbcbr1g  4687  csbfv12g  5336  isorel  5489  isocnv  5491  isotr  5495  caovord  5629  trtxp  5781  addcfnex  5824  funsex  5828  qrpprod  5836  enpw1  6062  enmap2  6068  enpw  6087  cenc  6181  leaddc1  6214  tlecg  6230  ce2le  6233  lemuc1  6253