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Theorem sbcbid 3829
Description: Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
sbcbid.1 𝑥𝜑
sbcbid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcbid (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Proof of Theorem sbcbid
StepHypRef Expression
1 sbcbid.1 . . . 4 𝑥𝜑
2 sbcbid.2 . . . 4 (𝜑 → (𝜓𝜒))
31, 2abbid 2891 . . 3 (𝜑 → {𝑥𝜓} = {𝑥𝜒})
43eleq2d 2902 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝐴 ∈ {𝑥𝜒}))
5 df-sbc 3776 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
6 df-sbc 3776 . 2 ([𝐴 / 𝑥]𝜒𝐴 ∈ {𝑥𝜒})
74, 5, 63bitr4g 315 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wnf 1777  wcel 2106  {cab 2802  [wsbc 3775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-sbc 3776
This theorem is referenced by:  sbcbidvOLD  3831  sbcbi2  3834  csbeq2d  3892
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