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Theorem sbcbr 4841
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 4840 . 2 ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
2 csbconstg 3695 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐵)
3 csbconstg 3695 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐶 = 𝐶)
42, 3breq12d 4799 . . 3 (𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐵𝐴 / 𝑥𝑅𝐶))
5 br0 4835 . . . . 5 ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶
6 csbprc 4124 . . . . . 6 𝐴 ∈ V → 𝐴 / 𝑥𝑅 = ∅)
76breqd 4797 . . . . 5 𝐴 ∈ V → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
85, 7mtbiri 316 . . . 4 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝑅𝐴 / 𝑥𝐶)
9 br0 4835 . . . . 5 ¬ 𝐵𝐶
106breqd 4797 . . . . 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 2145  Vcvv 3351  [wsbc 3587  csb 3682  c0 4063   class class class wbr 4786
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 827  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 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787
This theorem is referenced by:  csbcnv  5444
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