Metamath Proof Explorer

Theorem csbeq2gVD

Description: Virtual deduction proof of csbeq2 . 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. csbeq2 is csbeq2gVD without virtual deductions and was automatically derived from csbeq2gVD .

1:: |- (. A e. V ->. A e. V ).
2:1: |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ).
3:1: |- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ).
4:2,3: |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ).
qed:4: |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) )
(Contributed by Alan Sare, 10-Nov-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion csbeq2gVD ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝐵 = 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    𝐴 ∈ 𝑉    ▶    𝐴 ∈ 𝑉    )
2 spsbc ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝐵 = 𝐶 → [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ) )
3 1 2 e1a ⊢ (    𝐴 ∈ 𝑉    ▶    ( ∀ 𝑥 𝐵 = 𝐶 → [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 )    )
4 sbceqg ⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )
5 1 4 e1a ⊢ (    𝐴 ∈ 𝑉    ▶    ( [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )    )
6 imbi2 ⊢ ( ( [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) → ( ( ∀ 𝑥 𝐵 = 𝐶 → [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ) ↔ ( ∀ 𝑥 𝐵 = 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) ) )
7 6 biimpcd ⊢ ( ( ∀ 𝑥 𝐵 = 𝐶 → [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ) → ( ( [ 𝐴 / 𝑥 ] 𝐵 = 𝐶 ↔ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) → ( ∀ 𝑥 𝐵 = 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) ) )
8 3 5 7 e11 ⊢ (    𝐴 ∈ 𝑉    ▶    ( ∀ 𝑥 𝐵 = 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )    )
9 8 in1 ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝐵 = 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 ) )