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Theorem sbcbr 5198
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 5197 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
2 csbconstg 3918 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3918 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3breq12d 5156 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
5 br0 5192 . . . . 5 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
6 csbprc 4409 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
76breqd 5154 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
85, 7mtbiri 327 . . . 4 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
9 br0 5192 . . . . 5 ¬ 𝐵𝐶
106breqd 5154 . . . . 5 𝐴 ∈ V → (𝐵𝐴 / 𝑥𝑅𝐶𝐵𝐶))
119, 10mtbiri 327 . . . 4 𝐴 ∈ V → ¬ 𝐵𝐴 / 𝑥𝑅𝐶)
128, 112falsed 376 . . 3 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
134, 12pm2.61i 182 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶)
141, 13bitri 275 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wcel 2108  Vcvv 3480  [wsbc 3788  csb 3899  c0 4333   class class class wbr 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144
This theorem is referenced by:  csbcnv  5894
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