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Theorem sbcbr2g 5205
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr2g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑅
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem sbcbr2g
StepHypRef Expression
1 sbcbr12g 5203 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
2 csbconstg 3911 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐵)
32breq1d 5157 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶𝐵𝑅𝐴 / 𝑥𝐶))
41, 3bitrd 279 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝑅𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  [wsbc 3776  csb 3892   class class class wbr 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148
This theorem is referenced by:  prmgaplem7  16986  telgsums  19853  fvmptnn04if  22333  bnj110  33807  frege124d  42445  frege72  42619  frege91  42638  frege116  42663  frege120  42667
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