Proof of Theorem sbcrext
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sbcex 3798 | . . 3
⊢
([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V) | 
| 2 | 1 | a1i 11 | . 2
⊢
(Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)) | 
| 3 |  | nfnfc1 2908 | . . 3
⊢
Ⅎ𝑦Ⅎ𝑦𝐴 | 
| 4 |  | id 22 | . . . 4
⊢
(Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴) | 
| 5 |  | nfcvd 2906 | . . . 4
⊢
(Ⅎ𝑦𝐴 → Ⅎ𝑦V) | 
| 6 | 4, 5 | nfeld 2917 | . . 3
⊢
(Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V) | 
| 7 |  | sbcex 3798 | . . . 4
⊢
([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | 
| 8 | 7 | 2a1i 12 | . . 3
⊢
(Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))) | 
| 9 | 3, 6, 8 | rexlimd2 3265 | . 2
⊢
(Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)) | 
| 10 |  | sbcng 3836 | . . . . . 6
⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑)) | 
| 11 | 10 | adantl 481 | . . . . 5
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑)) | 
| 12 |  | sbcralt 3872 | . . . . . . . 8
⊢ ((𝐴 ∈ V ∧
Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑)) | 
| 13 | 12 | ancoms 458 | . . . . . . 7
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑)) | 
| 14 | 3, 6 | nfan1 2200 | . . . . . . . 8
⊢
Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) | 
| 15 |  | sbcng 3836 | . . . . . . . . 9
⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | 
| 16 | 15 | adantl 481 | . . . . . . . 8
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)) | 
| 17 | 14, 16 | ralbid 3273 | . . . . . . 7
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)) | 
| 18 | 13, 17 | bitrd 279 | . . . . . 6
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)) | 
| 19 | 18 | notbid 318 | . . . . 5
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)) | 
| 20 | 11, 19 | bitrd 279 | . . . 4
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)) | 
| 21 |  | dfrex2 3073 | . . . . 5
⊢
(∃𝑦 ∈
𝐵 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑) | 
| 22 | 21 | sbcbii 3846 | . . . 4
⊢
([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑) | 
| 23 |  | dfrex2 3073 | . . . 4
⊢
(∃𝑦 ∈
𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑) | 
| 24 | 20, 22, 23 | 3bitr4g 314 | . . 3
⊢
((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | 
| 25 | 24 | ex 412 | . 2
⊢
(Ⅎ𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))) | 
| 26 | 2, 9, 25 | pm5.21ndd 379 | 1
⊢
(Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |