MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcimdvOLD Structured version   Visualization version   GIF version

Theorem sbcimdvOLD 3792
Description: Obsolete version of sbcimdv 3791 as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sbcimdvOLD.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
sbcimdvOLD (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem sbcimdvOLD
StepHypRef Expression
1 sbcex 3727 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
2 sbcimdvOLD.1 . . . . 5 (𝜑 → (𝜓𝜒))
32alrimiv 1930 . . . 4 (𝜑 → ∀𝑥(𝜓𝜒))
4 spsbc 3730 . . . 4 (𝐴 ∈ V → (∀𝑥(𝜓𝜒) → [𝐴 / 𝑥](𝜓𝜒)))
5 sbcim1 3773 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
63, 4, 5syl56 36 . . 3 (𝐴 ∈ V → (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
76com3l 89 . 2 (𝜑 → ([𝐴 / 𝑥]𝜓 → (𝐴 ∈ V → [𝐴 / 𝑥]𝜒)))
81, 7mpdi 45 1 (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1537  wcel 2106  Vcvv 3431  [wsbc 3717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-sbc 3718
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator