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Theorem 19.41rgVD 44239
Description: Virtual deduction proof of 19.41rg 43887. 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 43887 is 19.41rgVD 44239 without virtual deductions and was automatically derived from 19.41rgVD 44239. (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 43911 . . . . . . . . 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 1789 . . . . . . . . . 10 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
6 al2im 1808 . . . . . . . . . 10 (∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓)))) → (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))))
75, 6e0a 44109 . . . . . . . . 9 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
81, 7e1a 43964 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
9 idn2 43950 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥𝜓   )
10 id 22 . . . . . . . 8 ((∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
118, 9, 10e12 44061 . . . . . . 7 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥(𝜑 → (𝜑𝜓))   )
12 exim 1828 . . . . . . 7 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
1311, 12e2 43968 . . . . . 6 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
1413in2 43942 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
15 sp 2168 . . . . . 6 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
161, 15e1a 43964 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀𝑥𝜓)   )
17 imim2 58 . . . . 5 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → ((𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))))
1814, 16, 17e11 44025 . . . 4 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
19 pm2.04 90 . . . 4 ((𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
2018, 19e1a 43964 . . 3 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
21 pm3.31 449 . . 3 ((∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
2220, 21e1a 43964 . 2 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
2322in1 43908 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-vd1 43907  df-vd2 43915
This theorem is referenced by: (None)
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