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Theorem 19.41rgVD 39637
Description: Virtual deduction proof of 19.41rg 39268. 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 39268 is 19.41rgVD 39637 without virtual deductions and was automatically derived from 19.41rgVD 39637. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) 2:1: ⊢ ((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → ( 𝜑 ∧ 𝜓)))) 3:2: ⊢ ∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))) 4:3: ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))) 5:: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ∀𝑥(𝜓 → ∀𝑥𝜓)   ) 6:4,5: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑 ∧ 𝜓)))   ) 7:: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥𝜓   ) 8:6,7: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    ∀𝑥(𝜑 → (𝜑 ∧ 𝜓))   ) 9:8: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ,   ∀𝑥𝜓   ▶    (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))   ) 10:9: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 11:5: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀ 𝑥𝜓)   ) 12:10,11: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ( ∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 13:12: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))   ) 14:13: ⊢ (   ∀𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥 𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))   ) qed:14: ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
Assertion
Ref Expression
19.41rgVD (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))

Proof of Theorem 19.41rgVD
StepHypRef Expression
1 idn1 39292 . . . . . . . . 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 1871 . . . . . . . . . 10 𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓))))
6 al2im 1891 . . . . . . . . . 10 (∀𝑥((𝜓 → ∀𝑥𝜓) → (𝜓 → (𝜑 → (𝜑𝜓)))) → (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))))
75, 6e0a 39501 . . . . . . . . 9 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
81, 7e1a 39354 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓)))   )
9 idn2 39340 . . . . . . . 8 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥𝜓   )
10 id 22 . . . . . . . 8 ((∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))) → (∀𝑥𝜓 → ∀𝑥(𝜑 → (𝜑𝜓))))
118, 9, 10e12 39453 . . . . . . 7 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   𝑥(𝜑 → (𝜑𝜓))   )
12 exim 1910 . . . . . . 7 (∀𝑥(𝜑 → (𝜑𝜓)) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
1311, 12e2 39358 . . . . . 6 (   𝑥(𝜓 → ∀𝑥𝜓)   ,   𝑥𝜓   ▶   (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))   )
1413in2 39332 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
15 sp 2200 . . . . . 6 (∀𝑥(𝜓 → ∀𝑥𝜓) → (𝜓 → ∀𝑥𝜓))
161, 15e1a 39354 . . . . 5 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → ∀𝑥𝜓)   )
17 imim2 58 . . . . 5 ((∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → ((𝜓 → ∀𝑥𝜓) → (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))))
1814, 16, 17e11 39415 . . . 4 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))   )
19 pm2.04 90 . . . 4 ((𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓))) → (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))))
2018, 19e1a 39354 . . 3 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   (∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓)))   )
21 pm3.31 460 . . 3 ((∃𝑥𝜑 → (𝜓 → ∃𝑥(𝜑𝜓))) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
2220, 21e1a 39354 . 2 (   𝑥(𝜓 → ∀𝑥𝜓)   ▶   ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓))   )
2322in1 39289 1 (∀𝑥(𝜓 → ∀𝑥𝜓) → ((∃𝑥𝜑𝜓) → ∃𝑥(𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1630  ∃wex 1853 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-vd1 39288  df-vd2 39296 This theorem is referenced by: (None)
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