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Theorem sbcbr1g 4844
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 4843 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
2 csbconstg 3695 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32breq2d 4799 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝑅𝐶))
41, 3bitrd 268 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2145  [wsbc 3587  csb 3682   class class class wbr 4787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788
This theorem is referenced by:  frege124d  38579  frege70  38753  frege92  38775  frege118  38801
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