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Theorem sbcbiVD 44309
Description: Implication form of sbcbii 3835. 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 43972 is sbcbiVD 44309 without virtual deductions and was automatically derived from sbcbiVD 44309.
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 44007 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 44046 . . . . 5 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   𝑥(𝜑𝜓)   )
3 spsbc 3788 . . . . 5 (𝐴𝐵 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓)))
41, 2, 3e12 44157 . . . 4 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   [𝐴 / 𝑥](𝜑𝜓)   )
5 sbcbig 3829 . . . . 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 44157 . . 3 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)   )
87in2 44038 . 2 (   𝐴𝐵   ▶   (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
98in1 44004 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 3775
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 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3776  df-vd1 44003  df-vd2 44011
This theorem is referenced by: (None)
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