# 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

### Proof

Step Hyp Ref Expression
1 idn2 ${⊢}\left({A}\in {B}{,}{C}\in {D}{\to }{C}\in {D}\right)$
2 idn1 ${⊢}\left({A}\in {B}{\to }{A}\in {B}\right)$
3 idn3 ${⊢}\left({A}\in {B}{,}{C}\in {D}{,}\forall {x}\in {B}\phantom{\rule{.4em}{0ex}}\forall {y}\in {D}\phantom{\rule{.4em}{0ex}}{\phi }{\to }\forall {x}\in {B}\phantom{\rule{.4em}{0ex}}\forall {y}\in {D}\phantom{\rule{.4em}{0ex}}{\phi }\right)$
4 rspsbc
5 2 3 4 e13
6 sbcralg
7 6 biimpd
8 2 5 7 e13
9 rspsbc
10 1 8 9 e23
11 10 in3
12 11 in2
13 12 in1