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Theorem 19.41rgVD 38160
Description: Virtual deduction proof of 19.41rg 37787. 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 37787 is 19.41rgVD 38160 without virtual deductions and was automatically derived from 19.41rgVD 38160. (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 37811 . . . . . . . . 9 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   𝑥(𝜓 → ∀𝑥𝜓)   )
2 pm3.2 462 . . . . . . . . . . . . 13 (𝜑 → (𝜓 → (𝜑𝜓)))
32com12 32 . . . . . . . . . . . 12 (𝜓 → (𝜑 → (𝜑𝜓)))
43a1i 11 . . . . . . . . . . 11 ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
54ax-gen 1713 . . . . . . . . . 10 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
6 al2im 1732 . . . . . . . . . 10 (∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓)))) → (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))))
75, 6e0a 38020 . . . . . . . . 9 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
81, 7e1a 37873 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
9 idn2 37859 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥𝜓   )
10 id 22 . . . . . . . 8 ((∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
118, 9, 10e12 37972 . . . . . . 7 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥(𝜑 → (𝜑𝜓))   )
12 exim 1751 . . . . . . 7 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
1311, 12e2 37877 . . . . . 6 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
1413in2 37851 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
15 sp 2041 . . . . . 6 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
161, 15e1a 37873 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀𝑥𝜓)   )
17 imim2 56 . . . . 5 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → ((𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))))
1814, 16, 17e11 37934 . . . 4 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
19 pm2.04 88 . . . 4 ((𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
2018, 19e1a 37873 . . 3 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
21 pm3.31 460 . . 3 ((∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
2220, 21e1a 37873 . 2 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
2322in1 37808 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wex 1695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-vd1 37807  df-vd2 37815
This theorem is referenced by: (None)
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