Metamath Proof Explorer

Theorem onfrALTlem4VD

Description: Virtual deduction proof of onfrALTlem4 . 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. onfrALTlem4 is onfrALTlem4VD without virtual deductions and was automatically derived from onfrALTlem4VD .

1:: |- y e.V
2:1: |- ( [. y / x ]. ( a i^i x ) = (/) <-> [ y / x ]_ ( a i^i x ) = [_ y / x ]_ (/) )
3:1: |- [_ y / x ]_ ( a i^i x ) = ( [_ y / x ]_ a i^i [_ y / x ]_ x )
4:1: |- [_ y / x ]_ a = a
5:1: |- [_ y / x ]_ x = y
6:4,5: |- ( [_ y / x ]_ a i^i [_ y / x ]_ x ) = ( a i^i y )
7:3,6: |- [_ y / x ]_ ( a i^i x ) = ( a i^i y )
8:1: |- [_ y / x ]_ (/) = (/)
9:7,8: |- ( [_ y / x ]_ ( a i^i x ) = [_ y / x ]_ (/) <-> ( a i^i y ) = (/) )
10:2,9: |- ( [. y / x ]. ( a i^i x ) = (/) <-> ( a i^i y ) = (/) )
11:1: |- ( [. y / x ]. x e. a <-> y e. a )
12:11,10: |- ( ( [. y / x ]. x e. a /\ [. y / x ]. ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
13:1: |- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( [. y / x ]. x e. a /\ [. y / x ]. ( a i^i x ) = (/) ) )
qed:13,12: |- ( [. y / x ]. ( x e. a /\ ( a i^i x ) = (/) ) <-> ( y e. a /\ ( a i^i y ) = (/) ) )
(Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion onfrALTlem4VD ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )

Proof

Step Hyp Ref Expression
1 sbcan ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ∧ [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ) )
2 sbcel1v ⊢ ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ↔ 𝑦 ∈ 𝑎 )
3 sbceqg ⊢ ( 𝑦 ∈ V → ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ ) )
4 3 elv ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ )
5 csbin ⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ( ⦋ 𝑦 / 𝑥 ⦌ 𝑎 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 )
6 csbconstg ⊢ ( 𝑦 ∈ V → ⦋ 𝑦 / 𝑥 ⦌ 𝑎 = 𝑎 )
7 6 elv ⊢ ⦋ 𝑦 / 𝑥 ⦌ 𝑎 = 𝑎
8 vex ⊢ 𝑦 ∈ V
9 8 csbvargi ⊢ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 = 𝑦
10 7 9 ineq12i ⊢ ( ⦋ 𝑦 / 𝑥 ⦌ 𝑎 ∩ ⦋ 𝑦 / 𝑥 ⦌ 𝑥 ) = ( 𝑎 ∩ 𝑦 )
11 5 10 eqtri ⊢ ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ( 𝑎 ∩ 𝑦 )
12 csb0 ⊢ ⦋ 𝑦 / 𝑥 ⦌ ∅ = ∅
13 11 12 eqeq12i ⊢ ( ⦋ 𝑦 / 𝑥 ⦌ ( 𝑎 ∩ 𝑥 ) = ⦋ 𝑦 / 𝑥 ⦌ ∅ ↔ ( 𝑎 ∩ 𝑦 ) = ∅ )
14 4 13 bitri ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ↔ ( 𝑎 ∩ 𝑦 ) = ∅ )
15 2 14 anbi12i ⊢ ( ( [ 𝑦 / 𝑥 ] 𝑥 ∈ 𝑎 ∧ [ 𝑦 / 𝑥 ] ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )
16 1 15 bitri ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑥 ) = ∅ ) ↔ ( 𝑦 ∈ 𝑎 ∧ ( 𝑎 ∩ 𝑦 ) = ∅ ) )