MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcbr1g Structured version   Visualization version   GIF version

Theorem sbcbr1g 5198
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr1g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑅
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcbr1g
StepHypRef Expression
1 sbcbr12g 5197 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
2 csbconstg 3907 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32breq2d 5153 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝑅𝐶))
41, 3bitrd 279 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  [wsbc 3772  csb 3888   class class class wbr 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142
This theorem is referenced by:  iscard4  42842  frege124d  43070  frege70  43242  frege92  43264  frege118  43290
  Copyright terms: Public domain W3C validator