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Theorem sbcimdv 2880
Description: Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1387). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
Hypothesis
Ref Expression
sbcimdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdv (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdv
StepHypRef Expression
1 sbcex 2824 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
2 sbcimdv.1 . . . . 5 (𝜑 → (𝜓𝜒))
32alrimiv 1796 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbc 2827 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
5 sbcim1 2863 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
63, 4, 5syl56 34 . . 3 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76com3l 80 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
81, 7mpdi 42 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wcel 1434  Vcvv 2602  [wsbc 2816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817
This theorem is referenced by: (None)
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