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Theorem ssralv2VD 39416
Description: Quantification restricted to a subclass for two quantifiers. ssralv 3699 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 39054 is ssralv2VD 39416 without virtual deductions and was automatically derived from ssralv2VD 39416.
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 1879 . . . . 5 ((𝐴𝐵𝐶𝐷) → ∀𝑥(𝐴𝐵𝐶𝐷))
2 hbra1 2971 . . . . 5 (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝑥𝐵𝑦𝐷 𝜑)
3 idn1 39107 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ▶   (𝐴𝐵𝐶𝐷)   )
4 simpr 476 . . . . . . . 8 ((𝐴𝐵𝐶𝐷) → 𝐶𝐷)
53, 4e1a 39169 . . . . . . 7 (   (𝐴𝐵𝐶𝐷)   ▶   𝐶𝐷   )
6 idn3 39157 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑥𝐴   )
7 simpl 472 . . . . . . . . . . . 12 ((𝐴𝐵𝐶𝐷) → 𝐴𝐵)
83, 7e1a 39169 . . . . . . . . . . 11 (   (𝐴𝐵𝐶𝐷)   ▶   𝐴𝐵   )
9 idn2 39155 . . . . . . . . . . 11 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐵𝑦𝐷 𝜑   )
10 ssralv 3699 . . . . . . . . . . 11 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
118, 9, 10e12 39268 . . . . . . . . . 10 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐴𝑦𝐷 𝜑   )
12 df-ral 2946 . . . . . . . . . . 11 (∀𝑥𝐴𝑦𝐷 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
1312biimpi 206 . . . . . . . . . 10 (∀𝑥𝐴𝑦𝐷 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
1411, 13e2 39173 . . . . . . . . 9 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑)   )
15 sp 2091 . . . . . . . . 9 (∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑) → (𝑥𝐴 → ∀𝑦𝐷 𝜑))
1614, 15e2 39173 . . . . . . . 8 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐷 𝜑)   )
17 pm2.27 42 . . . . . . . 8 (𝑥𝐴 → ((𝑥𝐴 → ∀𝑦𝐷 𝜑) → ∀𝑦𝐷 𝜑))
186, 16, 17e32 39302 . . . . . . 7 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑦𝐷 𝜑   )
19 ssralv 3699 . . . . . . 7 (𝐶𝐷 → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
205, 18, 19e13 39292 . . . . . 6 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ,   𝑥𝐴   ▶   𝑦𝐶 𝜑   )
2120in3 39151 . . . . 5 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   (𝑥𝐴 → ∀𝑦𝐶 𝜑)   )
221, 2, 21gen21nv 39162 . . . 4 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑)   )
23 df-ral 2946 . . . . 5 (∀𝑥𝐴𝑦𝐶 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑))
2423biimpri 218 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦𝐶 𝜑) → ∀𝑥𝐴𝑦𝐶 𝜑)
2522, 24e2 39173 . . 3 (   (𝐴𝐵𝐶𝐷)   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐴𝑦𝐶 𝜑   )
2625in2 39147 . 2 (   (𝐴𝐵𝐶𝐷)   ▶   (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑)   )
2726in1 39104 1 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wcel 2030  wral 2941  wss 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ral 2946  df-in 3614  df-ss 3621  df-vd1 39103  df-vd2 39111  df-vd3 39123
This theorem is referenced by: (None)
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