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Theorem 19.41rgVD 44927
Description: Virtual deduction proof of 19.41rg 44575. 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. 19.41rg 44575 is 19.41rgVD 44927 without virtual deductions and was automatically derived from 19.41rgVD 44927. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
1:: (𝜓 → (𝜑 → (𝜑𝜓)))
2:1: ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → ( 𝜑𝜓))))
3:2: 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
4:3: (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 𝑥(𝜑 → (𝜑𝜓))))
5:: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   𝑥(𝜓 → ∀𝑥𝜓)   )
6:4,5: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
7:: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    𝑥𝜓   )
8:6,7: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    𝑥(𝜑 → (𝜑𝜓))   )
9:8: (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶    (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
10:9: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
11:5: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀ 𝑥𝜓)   )
12:10,11: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ( 𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
13:12: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
14:13: (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥 𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
qed:14: (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 𝜓) → ∃𝑥(𝜑𝜓)))
Assertion
Ref Expression
19.41rgVD (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Proof of Theorem 19.41rgVD
StepHypRef Expression
1 idn1 44599 . . . . . . . . 9 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   𝑥(𝜓 → ∀𝑥𝜓)   )
2 pm3.2 469 . . . . . . . . . . . . 13 (𝜑 → (𝜓 → (𝜑𝜓)))
32com12 32 . . . . . . . . . . . 12 (𝜓 → (𝜑 → (𝜑𝜓)))
43a1i 11 . . . . . . . . . . 11 ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
54ax-gen 1794 . . . . . . . . . 10 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
6 al2im 1813 . . . . . . . . . 10 (∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓)))) → (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))))
75, 6e0a 44797 . . . . . . . . 9 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
81, 7e1a 44652 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
9 idn2 44638 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥𝜓   )
10 id 22 . . . . . . . 8 ((∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
118, 9, 10e12 44749 . . . . . . 7 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥(𝜑 → (𝜑𝜓))   )
12 exim 1833 . . . . . . 7 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
1311, 12e2 44656 . . . . . 6 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
1413in2 44630 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
15 sp 2182 . . . . . 6 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
161, 15e1a 44652 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀𝑥𝜓)   )
17 imim2 58 . . . . 5 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → ((𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))))
1814, 16, 17e11 44713 . . . 4 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
19 pm2.04 90 . . . 4 ((𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
2018, 19e1a 44652 . . 3 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
21 pm3.31 449 . . 3 ((∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
2220, 21e1a 44652 . 2 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
2322in1 44596 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-vd1 44595  df-vd2 44603
This theorem is referenced by: (None)
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