Metamath Proof Explorer

Theorem ssralv2VD

Description: Quantification restricted to a subclass for two quantifiers. ssralv for two quantifiers. 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. ssralv2 is ssralv2VD without virtual deductions and was automatically derived from ssralv2VD .

1:: |- (. ( A C_ B /\ C C_ D ) ->. ( A C_ B /\ C C_ D ) ).
2:: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. B A. y e. D ph ).
3:1: |- (. ( A C_ B /\ C C_ D ) ->. A C_ B ).
4:3,2: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. D ph ).
5:4: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. D ph ) ).
6:5: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. D ph ) ).
7:: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. x e. A ).
8:7,6: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. D ph ).
9:1: |- (. ( A C_ B /\ C C_ D ) ->. C C_ D ).
10:9,8: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph , x e. A ->. A. y e. C ph ).
11:10: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. ( x e. A -> A. y e. C ph ) ).
12:: |- ( ( A C_ B /\ C C_ D ) -> A. x ( A C_ B /\ C C_ D ) )
13:: |- ( A. x e. B A. y e. D ph -> A. x A. x e. B A. y e. D ph )
14:12,13,11: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x ( x e. A -> A. y e. C ph ) ).
15:14: |- (. ( A C_ B /\ C C_ D ) ,. A. x e. B A. y e. D ph ->. A. x e. A A. y e. C ph ).
16:15: |- (. ( A C_ B /\ C C_ D ) ->. ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) ).
qed:16: |- ( ( A C_ B /\ C C_ D ) -> ( A. x e. B A. y e. D ph -> A. x e. A A. y e. C ph ) )
(Contributed by Alan Sare, 10-Feb-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion ssralv2VD ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ) )

Proof

Step Hyp Ref Expression
1 ax-5 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ∀ 𝑥 ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) )
2 hbra1 ⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 )
3 idn1 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ▶    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    )
4 simpr ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → 𝐶 ⊆ 𝐷 )
5 3 4 e1a ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ▶    𝐶 ⊆ 𝐷    )
6 idn3 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ,    𝑥 ∈ 𝐴    ▶    𝑥 ∈ 𝐴    )
7 simpl ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → 𝐴 ⊆ 𝐵 )
8 3 7 e1a ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ▶    𝐴 ⊆ 𝐵    )
9 idn2 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    )
10 ssralv ⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 ) )
11 8 9 10 e12 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑    )
12 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
13 12 biimpi ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
14 11 13 e2 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 )    )
15 sp ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) → ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
16 14 15 e2 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 )    )
17 pm2.27 ⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐷 𝜑 ) → ∀ 𝑦 ∈ 𝐷 𝜑 ) )
18 6 16 17 e32 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ,    𝑥 ∈ 𝐴    ▶    ∀ 𝑦 ∈ 𝐷 𝜑    )
19 ssralv ⊢ ( 𝐶 ⊆ 𝐷 → ( ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
20 5 18 19 e13 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ,    𝑥 ∈ 𝐴    ▶    ∀ 𝑦 ∈ 𝐶 𝜑    )
21 20 in3 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 )    )
22 1 2 21 gen21nv ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 )    )
23 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) )
24 23 biimpri ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐶 𝜑 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 )
25 22 24 e2 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ,    ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑    ▶    ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑    )
26 25 in2 ⊢ (    ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 )    ▶    ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 )    )
27 26 in1 ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐷 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝜑 ) )