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Theorem sbcbr 4637
 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 4636 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
2 csbconstg 3512 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3512 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3breq12d 4596 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
5 br0 4631 . . . . 5 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
6 csbprc 3932 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
76breqd 4594 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
85, 7mtbiri 316 . . . 4 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
9 br0 4631 . . . . 5 ¬ 𝐵𝐶
106breqd 4594 . . . . 5 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝑅𝐶𝐵𝐶))
119, 10mtbiri 316 . . . 4 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝑅𝐶)
128, 112falsed 365 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
134, 12pm2.61i 175 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶)
141, 13bitri 263 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by:  csbcnv  5228
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