Metamath Proof Explorer

Theorem 19.41rgVD

Description: Virtual deduction proof of 19.41rg . 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 is 19.41rgVD without virtual deductions and was automatically derived from 19.41rgVD . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

1:: |- ( ps -> ( ph -> ( ph /\ ps ) ) )
2:1: |- ( ( ps -> A. x ps ) -> ( ps -> ( ph -> ( ph /\ ps ) ) ) )
3:2: |- A. x ( ( ps -> A. x ps ) -> ( ps -> ( ph -> ( ph /\ ps ) ) ) )
4:3: |- ( A. x ( ps -> A. x ps ) -> ( A. x ps -> A. x ( ph -> ( ph /\ ps ) ) ) )
5:: |- (. A. x ( ps -> A. x ps ) ->. A. x ( ps -> A. x ps ) ).
6:4,5: |- (. A. x ( ps -> A. x ps ) ->. ( A. x ps -> A. x ( ph -> ( ph /\ ps ) ) ) ).
7:: |- (. A. x ( ps -> A. x ps ) ,. A. x ps ->. A. x ps ).
8:6,7: |- (. A. x ( ps -> A. x ps ) ,. A. x ps ->. A. x ( ph -> ( ph /\ ps ) ) ).
9:8: |- (. A. x ( ps -> A. x ps ) ,. A. x ps ->. ( E. x ph -> E. x ( ph /\ ps ) ) ).
10:9: |- (. A. x ( ps -> A. x ps ) ->. ( A. x ps -> ( E. x ph -> E. x ( ph /\ ps ) ) ) ).
11:5: |- (. A. x ( ps -> A. x ps ) ->. ( ps -> A. x ps ) ).
12:10,11: |- (. A. x ( ps -> A. x ps ) ->. ( ps -> ( E. x ph -> E. x ( ph /\ ps ) ) ) ).
13:12: |- (. A. x ( ps -> A. x ps ) ->. ( E. x ph -> ( ps -> E. x ( ph /\ ps ) ) ) ).
14:13: |- (. A. x ( ps -> A. x ps ) ->. ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) ) ).
qed:14: |- ( A. x ( ps -> A. x ps ) -> ( ( E. x ph /\ ps ) -> E. x ( ph /\ ps ) ) )

Ref Expression
Assertion 19.41rgVD ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    )
2 pm3.2 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) )
3 2 com12 ⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
4 3 a1i ⊢ ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
5 4 ax-gen ⊢ ∀ 𝑥 ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
6 al2im ⊢ ( ∀ 𝑥 ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) ) → ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) ) )
7 5 6 e0a ⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
8 1 7 e1a ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )    )
9 idn2 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ,    ∀ 𝑥 𝜓    ▶    ∀ 𝑥 𝜓    )
10 id ⊢ ( ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) ) )
11 8 9 10 e12 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ,    ∀ 𝑥 𝜓    ▶    ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) )    )
12 exim ⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
13 11 12 e2 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ,    ∀ 𝑥 𝜓    ▶    ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )    )
14 13 in2 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )    )
15 sp ⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ∀ 𝑥 𝜓 ) )
16 1 15 e1a ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( 𝜓 → ∀ 𝑥 𝜓 )    )
17 imim2 ⊢ ( ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) → ( ( 𝜓 → ∀ 𝑥 𝜓 ) → ( 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) ) )
18 14 16 17 e11 ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )    )
19 pm2.04 ⊢ ( ( 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) → ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) )
20 18 19 e1a ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )    )
21 pm3.31 ⊢ ( ( ∃ 𝑥 𝜑 → ( 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ) → ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
22 20 21 e1a ⊢ (    ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 )    ▶    ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )    )
23 22 in1 ⊢ ( ∀ 𝑥 ( 𝜓 → ∀ 𝑥 𝜓 ) → ( ( ∃ 𝑥 𝜑 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )