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 ) ) )