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Theorem sbcbiVD 44587
Description: Implication form of sbcbii 3837. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sbcbi 44250 is sbcbiVD 44587 without virtual deductions and was automatically derived from sbcbiVD 44587.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   𝑥(𝜑𝜓)   )
3:1,2: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   [𝐴 / 𝑥](𝜑𝜓)   )
4:1,3: (   𝐴𝐵   ,   𝑥(𝜑𝜓)    ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)   )
5:4: (   𝐴𝐵   ▶   (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
qed:5: (𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbcbiVD (𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcbiVD
StepHypRef Expression
1 idn1 44285 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 44324 . . . . 5 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   𝑥(𝜑𝜓)   )
3 spsbc 3789 . . . . 5 (𝐴𝐵 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓)))
41, 2, 3e12 44435 . . . 4 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   [𝐴 / 𝑥](𝜑𝜓)   )
5 sbcbig 3831 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
65biimpd 228 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
71, 4, 6e12 44435 . . 3 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)   )
87in2 44316 . 2 (   𝐴𝐵   ▶   (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
98in1 44282 1 (𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wcel 2099  [wsbc 3776
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3777  df-vd1 44281  df-vd2 44289
This theorem is referenced by: (None)
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