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Theorem 19.41rgVD 44406
Description: Virtual deduction proof of 19.41rg 44054. 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 44054 is 19.41rgVD 44406 without virtual deductions and was automatically derived from 19.41rgVD 44406. (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 44078 . . . . . . . . 9 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   𝑥(𝜓 → ∀𝑥𝜓)   )
2 pm3.2 468 . . . . . . . . . . . . 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 44276 . . . . . . . . 9 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
81, 7e1a 44131 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
9 idn2 44117 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥𝜓   )
10 id 22 . . . . . . . 8 ((∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
118, 9, 10e12 44228 . . . . . . 7 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥(𝜑 → (𝜑𝜓))   )
12 exim 1828 . . . . . . 7 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
1311, 12e2 44135 . . . . . 6 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
1413in2 44109 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
15 sp 2171 . . . . . 6 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
161, 15e1a 44131 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀𝑥𝜓)   )
17 imim2 58 . . . . 5 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → ((𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))))
1814, 16, 17e11 44192 . . . 4 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
19 pm2.04 90 . . . 4 ((𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
2018, 19e1a 44131 . . 3 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
21 pm3.31 448 . . 3 ((∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
2220, 21e1a 44131 . 2 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
2322in1 44075 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  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 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-vd1 44074  df-vd2 44082
This theorem is referenced by: (None)
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