Metamath Proof Explorer

Theorem hbimpgVD

Description: Virtual deduction proof of hbimpg . 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. hbimpg is hbimpgVD without virtual deductions and was automatically derived from hbimpgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

1:: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ).
2:1: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. A. x ( ph -> A. x ph ) ).
3:: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) , -. ph ->. -. ph ).
4:2: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. A. x ( -. ph -> A. x -. ph ) ).
5:4: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( -. ph -> A. x -. ph ) ).
6:3,5: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) , -. ph ->. A. x -. ph ).
7:: |- ( -. ph -> ( ph -> ps ) )
8:7: |- ( A. x -. ph -> A. x ( ph -> ps ) )
9:6,8: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) , -. ph ->. A. x ( ph -> ps ) ).
10:9: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( -. ph -> A. x ( ph -> ps ) ) ).
11:: |- ( ps -> ( ph -> ps ) )
12:11: |- ( A. x ps -> A. x ( ph -> ps ) )
13:1: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. A. x ( ps -> A. x ps ) ).
14:13: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( ps -> A. x ps ) ).
15:14,12: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( ps -> A. x ( ph -> ps ) ) ).
16:10,15: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( ( -. ph \/ ps ) -> A. x ( ph -> ps ) ) ).
17:: |- ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
18:16,17: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. ( ( ph -> ps ) -> A. x ( ph -> ps ) ) ).
19:: |- ( A. x ( ph -> A. x ph ) -> A. x A. x ( ph -> A. x ph ) )
20:: |- ( A. x ( ps -> A. x ps ) -> A. x A. x ( ps -> A. x ps ) )
21:19,20: |- ( ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) -> A. x ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) )
22:21,18: |- (. ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) ->. A. x ( ( ph -> ps ) -> A. x ( ph -> ps ) ) ).
qed:22: |- ( ( A. x ( ph -> A. x ph ) /\ A. x ( ps -> A. x ps ) ) -> A. x ( ( ph -> ps ) -> A. x ( ph -> ps ) ) )

Ref Expression
Assertion hbimpgVD ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ψ → ∀ x φ → ψ

Proof

Step Hyp Ref Expression
1 hba1 ⊢ ∀ x φ → ∀ x φ → ∀ x ∀ x φ → ∀ x φ
2 hba1 ⊢ ∀ x ψ → ∀ x ψ → ∀ x ∀ x ψ → ∀ x ψ
3 1 2 hban ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ
4 idn2 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ , ¬ φ → ¬ φ
5 idn1 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ
6 simpl ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ∀ x φ
7 5 6 e1a ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ∀ x φ
8 hbntal ⊢ ∀ x φ → ∀ x φ → ∀ x ¬ φ → ∀ x ¬ φ
9 7 8 e1a ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x ¬ φ → ∀ x ¬ φ
10 sp ⊢ ∀ x ¬ φ → ∀ x ¬ φ → ¬ φ → ∀ x ¬ φ
11 9 10 e1a ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ¬ φ → ∀ x ¬ φ
12 pm2.27 ⊢ ¬ φ → ¬ φ → ∀ x ¬ φ → ∀ x ¬ φ
13 4 11 12 e21 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ , ¬ φ → ∀ x ¬ φ
14 pm2.21 ⊢ ¬ φ → φ → ψ
15 14 alimi ⊢ ∀ x ¬ φ → ∀ x φ → ψ
16 13 15 e2 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ , ¬ φ → ∀ x φ → ψ
17 16 in2 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ¬ φ → ∀ x φ → ψ
18 simpr ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x ψ → ∀ x ψ
19 5 18 e1a ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x ψ → ∀ x ψ
20 sp ⊢ ∀ x ψ → ∀ x ψ → ψ → ∀ x ψ
21 19 20 e1a ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ψ → ∀ x ψ
22 ax-1 ⊢ ψ → φ → ψ
23 22 alimi ⊢ ∀ x ψ → ∀ x φ → ψ
24 imim1 ⊢ ψ → ∀ x ψ → ∀ x ψ → ∀ x φ → ψ → ψ → ∀ x φ → ψ
25 21 23 24 e10 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ψ → ∀ x φ → ψ
26 jao ⊢ ¬ φ → ∀ x φ → ψ → ψ → ∀ x φ → ψ → ¬ φ ∨ ψ → ∀ x φ → ψ
27 17 25 26 e11 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ¬ φ ∨ ψ → ∀ x φ → ψ
28 imor ⊢ φ → ψ ↔ ¬ φ ∨ ψ
29 imbi1 ⊢ φ → ψ ↔ ¬ φ ∨ ψ → φ → ψ → ∀ x φ → ψ ↔ ¬ φ ∨ ψ → ∀ x φ → ψ
30 29 biimprcd ⊢ ¬ φ ∨ ψ → ∀ x φ → ψ → φ → ψ ↔ ¬ φ ∨ ψ → φ → ψ → ∀ x φ → ψ
31 27 28 30 e10 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → φ → ψ → ∀ x φ → ψ
32 3 31 gen11nv ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ψ → ∀ x φ → ψ
33 32 in1 ⊢ ∀ x φ → ∀ x φ ∧ ∀ x ψ → ∀ x ψ → ∀ x φ → ψ → ∀ x φ → ψ