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Theorem sbcbr1g 5087
Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.)
Assertion
Ref Expression
sbcbr1g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑅
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcbr1g
StepHypRef Expression
1 sbcbr12g 5086 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶))
2 csbconstg 3809 . . 3 (𝐴𝑉𝐴 / 𝑥𝐶 = 𝐶)
32breq2d 5042 . 2 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝑅𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵𝑅𝐶))
41, 3bitrd 282 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝑅𝐶𝐴 / 𝑥𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2114  [wsbc 3680  csb 3790   class class class wbr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031
This theorem is referenced by:  iscard4  40694  frege124d  40915  frege70  41087  frege92  41109  frege118  41135
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