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Theorem sbcbr 4740
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 4739 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
2 csbconstg 3579 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3579 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3breq12d 4698 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
5 br0 4734 . . . . 5 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
6 csbprc 4013 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
76breqd 4696 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
85, 7mtbiri 316 . . . 4 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
9 br0 4734 . . . . 5 ¬ 𝐵𝐶
106breqd 4696 . . . . 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 176 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶)
141, 13bitri 264 1 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐵𝐴 / 𝑥𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wcel 2030  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948   class class class wbr 4685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  csbcnv  5338
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