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Theorem bj-sbel1 36848
Description: Version of sbcel1g 4421 when substituting a set. (Note: one could have a corresponding version of sbcel12 4416 when substituting a set, but the point here is that the antecedent of sbcel1g 4421 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 3795 . 2 ([𝑦 / 𝑥]𝐴𝐵[𝑦 / 𝑥]𝐴𝐵)
2 sbcel1g 4421 . . 3 (𝑦 ∈ V → ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵))
32elv 3482 . 2 ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
41, 3bitri 275 1 ([𝑦 / 𝑥]𝐴𝐵𝑦 / 𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2060  wcel 2104  Vcvv 3477  [wsbc 3791  csb 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-nul 4340
This theorem is referenced by:  bj-snsetex  36906
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