Proof of Theorem sbciegftOLD
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sbc5 3816 | . . 3
⊢
([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) | 
| 2 |  | biimp 215 | . . . . . . . 8
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | 
| 3 | 2 | imim2i 16 | . . . . . . 7
⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓))) | 
| 4 | 3 | impd 410 | . . . . . 6
⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) | 
| 5 | 4 | alimi 1811 | . . . . 5
⊢
(∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) | 
| 6 |  | 19.23t 2210 | . . . . . 6
⊢
(Ⅎ𝑥𝜓 → (∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓))) | 
| 7 | 6 | biimpa 476 | . . . . 5
⊢
((Ⅎ𝑥𝜓 ∧ ∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) | 
| 8 | 5, 7 | sylan2 593 | . . . 4
⊢
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) | 
| 9 | 8 | 3adant1 1131 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) | 
| 10 | 1, 9 | biimtrid 242 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 → 𝜓)) | 
| 11 |  | biimpr 220 | . . . . . . . 8
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | 
| 12 | 11 | imim2i 16 | . . . . . . 7
⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑))) | 
| 13 | 12 | com23 86 | . . . . . 6
⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑))) | 
| 14 | 13 | alimi 1811 | . . . . 5
⊢
(∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑))) | 
| 15 |  | 19.21t 2206 | . . . . . 6
⊢
(Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))) | 
| 16 | 15 | biimpa 476 | . . . . 5
⊢
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))) | 
| 17 | 14, 16 | sylan2 593 | . . . 4
⊢
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))) | 
| 18 | 17 | 3adant1 1131 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))) | 
| 19 |  | sbc6g 3818 | . . . 4
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | 
| 20 | 19 | 3ad2ant1 1134 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))) | 
| 21 | 18, 20 | sylibrd 259 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → [𝐴 / 𝑥]𝜑)) | 
| 22 | 10, 21 | impbid 212 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |