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Theorem bj-sbel1 35185
Description: Version of sbcel1g 4360 when substituting a set. (Note: one could have a corresponding version of sbcel12 4355 when substituting a set, but the point here is that the antecedent of sbcel1g 4360 is not needed when substituting a set.) (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sbel1 ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem bj-sbel1
StepHypRef Expression
1 sbsbc 3731 . 2 ([𝑦 / 𝑥]𝐴𝐵[𝑦 / 𝑥]𝐴𝐵)
2 sbcel1g 4360 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵))
32elv 3447 . 2 ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
41, 3bitri 274 1 ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2066  wcel 2105  Vcvv 3441  [wsbc 3727  csb 3843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-nul 4270
This theorem is referenced by:  bj-snsetex  35247
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