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∈Dφ→[˙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∈Dφ→∀x∈B∀y∈Dφ
4 rspsbc ⊢A∈B→∀x∈B∀y∈Dφ→[˙A/x]˙∀y∈Dφ
5 2 3 4 e13 ⊢A∈B,C∈D,∀x∈B∀y∈Dφ→[˙A/x]˙∀y∈Dφ
6 sbcralg ⊢A∈B→[˙A/x]˙∀y∈Dφ↔∀y∈D[˙A/x]˙φ
7 6 biimpd ⊢A∈B→[˙A/x]˙∀y∈Dφ→∀y∈D[˙A/x]˙φ
8 2 5 7 e13 ⊢A∈B,C∈D,∀x∈B∀y∈Dφ→∀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∈Dφ→[˙C/y]˙[˙A/x]˙φ
11 10 in3 ⊢A∈B,C∈D→∀x∈B∀y∈Dφ→[˙C/y]˙[˙A/x]˙φ
12 11 in2 ⊢A∈B→C∈D→∀x∈B∀y∈Dφ→[˙C/y]˙[˙A/x]˙φ
13 12 in1 ⊢A∈B→C∈D→∀x∈B∀y∈Dφ→[˙C/y]˙[˙A/x]˙φ