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Theorem sbcbr 4898
Description: Move substitution in and out of a binary relation. (Contributed by NM, 23-Aug-2018.)
Assertion
Ref Expression
sbcbr ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)

Proof of Theorem sbcbr
StepHypRef Expression
1 sbcbr123 4897 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
2 csbconstg 3741 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3741 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3breq12d 4856 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
5 br0 4892 . . . . 5 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
6 csbprc 4176 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
76breqd 4854 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
85, 7mtbiri 319 . . . 4 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
9 br0 4892 . . . . 5 ¬ 𝐵𝐶
106breqd 4854 . . . . 5 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝑅𝐶𝐵𝐶))
119, 10mtbiri 319 . . . 4 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝑅𝐶)
128, 112falsed 368 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
134, 12pm2.61i 177 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶)
141, 13bitri 267 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wcel 2157  Vcvv 3385  [wsbc 3633  csb 3728  c0 4115   class class class wbr 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844
This theorem is referenced by:  csbcnv  5509
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