Metamath Proof Explorer

Theorem rspsbc2VD

Description: Virtual deduction proof of rspsbc2 . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1:: |- (. A e. B ->. A e. B ).
2:: |- (. A e. B ,. C e. D ->. C e. D ).
3:: |- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. A. x e. B A. y e. D ph ).
4:1,3,?: e13 |- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. [. A / x ]. A. y e. D ph ).
5:1,4,?: e13 |- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. A. y e. D [. A / x ]. ph ).
6:2,5,?: e23 |- (. A e. B ,. C e. D ,. A. x e. B A. y e. D ph ->. [. C / y ]. [. A / x ]. ph ).
7:6: |- (. A e. B ,. C e. D ->. ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ).
8:7: |- (. A e. B ->. ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) ).
qed:8: |- ( A e. B -> ( C e. D -> ( A. x e. B A. y e. D ph -> [. C / y ]. [. A / x ]. ph ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion rspsbc2VD ⊢ A ∈ B → C ∈ D → ∀ x ∈ B ∀ y ∈ D φ → [˙C / y]˙ [˙A / x]˙ φ

Proof

Step Hyp Ref Expression
1 idn2 ⊢ A ∈ B , C ∈ D → C ∈ D
2 idn1 ⊢ A ∈ B → A ∈ B
3 idn3 ⊢ A ∈ B , C ∈ D , ∀ x ∈ B ∀ y ∈ D φ → ∀ x ∈ B ∀ y ∈ D φ
4 rspsbc ⊢ A ∈ B → ∀ x ∈ B ∀ y ∈ D φ → [˙A / x]˙ ∀ y ∈ D φ
5 2 3 4 e13 ⊢ A ∈ B , C ∈ D , ∀ x ∈ B ∀ y ∈ D φ → [˙A / x]˙ ∀ y ∈ D φ
6 sbcralg ⊢ A ∈ B → [˙A / x]˙ ∀ y ∈ D φ ↔ ∀ y ∈ D [˙A / x]˙ φ
7 6 biimpd ⊢ A ∈ B → [˙A / x]˙ ∀ y ∈ D φ → ∀ y ∈ D [˙A / x]˙ φ
8 2 5 7 e13 ⊢ A ∈ B , C ∈ D , ∀ x ∈ B ∀ y ∈ D φ → ∀ y ∈ D [˙A / x]˙ φ
9 rspsbc ⊢ C ∈ D → ∀ y ∈ D [˙A / x]˙ φ → [˙C / y]˙ [˙A / x]˙ φ
10 1 8 9 e23 ⊢ A ∈ B , C ∈ D , ∀ x ∈ B ∀ y ∈ D φ → [˙C / y]˙ [˙A / x]˙ φ
11 10 in3 ⊢ A ∈ B , C ∈ D → ∀ x ∈ B ∀ y ∈ D φ → [˙C / y]˙ [˙A / x]˙ φ
12 11 in2 ⊢ A ∈ B → C ∈ D → ∀ x ∈ B ∀ y ∈ D φ → [˙C / y]˙ [˙A / x]˙ φ
13 12 in1 ⊢ A ∈ B → C ∈ D → ∀ x ∈ B ∀ y ∈ D φ → [˙C / y]˙ [˙A / x]˙ φ