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Theorem ssralv2VD 39851
Description: Quantification restricted to a subclass for two quantifiers. ssralv 3861 for two quantifiers. 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. ssralv2 39506 is ssralv2VD 39851 without virtual deductions and was automatically derived from ssralv2VD 39851.
1:: (   (𝐴𝐵𝐶𝐷)   ▶   (𝐴𝐵 𝐶𝐷)   )
2:: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐵𝑦𝐷𝜑   )
3:1: (   (𝐴𝐵𝐶𝐷)   ▶   𝐴𝐵   )
4:3,2: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐴𝑦𝐷𝜑   )
5:4: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐷𝜑)   )
6:5: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐷𝜑)   )
7:: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑥𝐴   )
8:7,6: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑦𝐷𝜑   )
9:1: (   (𝐴𝐵𝐶𝐷)   ▶   𝐶𝐷   )
10:9,8: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑, 𝑥𝐴   ▶   𝑦𝐶𝜑   )
11:10: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐶𝜑)   )
12:: ((𝐴𝐵𝐶𝐷) → ∀𝑥(𝐴𝐵𝐶𝐷))
13:: (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝑥𝐵𝑦𝐷𝜑)
14:12,13,11: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐶𝜑)   )
15:14: (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐴𝑦𝐶𝜑   )
16:15: (   (𝐴𝐵𝐶𝐷)    ▶   (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝐴𝑦𝐶𝜑)   )
qed:16: ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷𝜑 → ∀𝑥𝐴𝑦𝐶𝜑))
(Contributed by Alan Sare, 10-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ssralv2VD ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑦,𝐶   𝑥,𝐷   𝑦,𝐷
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem ssralv2VD
StepHypRef Expression
1 ax-5 2006 . . . . 5 ((𝐴𝐵𝐶𝐷) → ∀𝑥(𝐴𝐵𝐶𝐷))
2 hbra1 3122 . . . . 5 (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝑥𝐵𝑦𝐷 𝜑)
3 idn1 39549 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ▶   (𝐴𝐵𝐶𝐷)   )
4 simpr 478 . . . . . . . 8 ((𝐴𝐵𝐶𝐷) → 𝐶𝐷)
53, 4e1a 39611 . . . . . . 7 (   (𝐴𝐵𝐶𝐷)   ▶   𝐶𝐷   )
6 idn3 39599 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑥𝐴   )
7 simpl 475 . . . . . . . . . . . 12 ((𝐴𝐵𝐶𝐷) → 𝐴𝐵)
83, 7e1a 39611 . . . . . . . . . . 11 (   (𝐴𝐵𝐶𝐷)   ▶   𝐴𝐵   )
9 idn2 39597 . . . . . . . . . . 11 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐵𝑦𝐷 𝜑   )
10 ssralv 3861 . . . . . . . . . . 11 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
118, 9, 10e12 39709 . . . . . . . . . 10 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐴𝑦𝐷 𝜑   )
12 df-ral 3093 . . . . . . . . . . 11 (∀𝑥𝐴𝑦𝐷 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
1312biimpi 208 . . . . . . . . . 10 (∀𝑥𝐴𝑦𝐷 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
1411, 13e2 39615 . . . . . . . . 9 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑)   )
15 sp 2217 . . . . . . . . 9 (∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑) → (𝑥𝐴 → ∀𝑦𝐷 𝜑))
1614, 15e2 39615 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐷 𝜑)   )
17 pm2.27 42 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝐴 → ∀𝑦𝐷 𝜑) → ∀𝑦𝐷 𝜑))
186, 16, 17e32 39743 . . . . . . 7 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑦𝐷 𝜑   )
19 ssralv 3861 . . . . . . 7 (𝐶𝐷 → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
205, 18, 19e13 39733 . . . . . 6 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑦𝐶 𝜑   )
2120in3 39593 . . . . 5 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐶 𝜑)   )
221, 2, 21gen21nv 39604 . . . 4 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑)   )
23 df-ral 3093 . . . . 5 (∀𝑥𝐴𝑦𝐶 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑))
2423biimpri 220 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑) → ∀𝑥𝐴𝑦𝐶 𝜑)
2522, 24e2 39615 . . 3 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐴𝑦𝐶 𝜑   )
2625in2 39589 . 2 (   (𝐴𝐵𝐶𝐷)   ▶   (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑)   )
2726in1 39546 1 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651  wcel 2157  wral 3088  wss 3768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-ral 3093  df-in 3775  df-ss 3782  df-vd1 39545  df-vd2 39553  df-vd3 39565
This theorem is referenced by: (None)
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