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Theorem sbcbiVD 39603
Description: Implication form of sbcbiiOLD 39235. 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 39243 is sbcbiVD 39603 without virtual deductions and was automatically derived from sbcbiVD 39603.
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 39284 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
2 idn2 39332 . . . . 5 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   𝑥(𝜑𝜓)   )
3 spsbc 3581 . . . . 5 (𝐴𝐵 → (∀𝑥(𝜑𝜓) → [𝐴 / 𝑥](𝜑𝜓)))
41, 2, 3e12 39445 . . . 4 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   [𝐴 / 𝑥](𝜑𝜓)   )
5 sbcbig 3613 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
65biimpd 219 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
71, 4, 6e12 39445 . . 3 (   𝐴𝐵   ,   𝑥(𝜑𝜓)   ▶   ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)   )
87in2 39324 . 2 (   𝐴𝐵   ▶   (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))   )
98in1 39281 1 (𝐴𝐵 → (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1622  wcel 2131  [wsbc 3568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-v 3334  df-sbc 3569  df-vd1 39280  df-vd2 39288
This theorem is referenced by: (None)
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