| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfsbc1v 3807 | . . 3
⊢
Ⅎ𝑥[𝑐 / 𝑥]𝜑 | 
| 2 |  | nfsbc1v 3807 | . . 3
⊢
Ⅎ𝑥[𝑤 / 𝑥]𝜑 | 
| 3 |  | sbceq1a 3798 | . . 3
⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑)) | 
| 4 |  | dfsbcq 3789 | . . 3
⊢ (𝑤 = 𝑐 → ([𝑤 / 𝑥]𝜑 ↔ [𝑐 / 𝑥]𝜑)) | 
| 5 | 1, 2, 3, 4 | reu8nf 3876 | . 2
⊢
(∃!𝑥 ∈
{𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐))) | 
| 6 |  | nfv 1913 | . . . . 5
⊢
Ⅎ𝑥𝜓 | 
| 7 |  | nfcv 2904 | . . . . . 6
⊢
Ⅎ𝑥{𝐴, 𝐵} | 
| 8 |  | nfv 1913 | . . . . . . 7
⊢
Ⅎ𝑥 𝐴 = 𝑐 | 
| 9 | 1, 8 | nfim 1895 | . . . . . 6
⊢
Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) | 
| 10 | 7, 9 | nfralw 3310 | . . . . 5
⊢
Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) | 
| 11 | 6, 10 | nfan 1898 | . . . 4
⊢
Ⅎ𝑥(𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) | 
| 12 |  | nfv 1913 | . . . . 5
⊢
Ⅎ𝑥𝜒 | 
| 13 |  | nfv 1913 | . . . . . . 7
⊢
Ⅎ𝑥 𝐵 = 𝑐 | 
| 14 | 1, 13 | nfim 1895 | . . . . . 6
⊢
Ⅎ𝑥([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) | 
| 15 | 7, 14 | nfralw 3310 | . . . . 5
⊢
Ⅎ𝑥∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) | 
| 16 | 12, 15 | nfan 1898 | . . . 4
⊢
Ⅎ𝑥(𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) | 
| 17 |  | reuprg.1 | . . . . 5
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | 
| 18 |  | eqeq1 2740 | . . . . . . 7
⊢ (𝑥 = 𝐴 → (𝑥 = 𝑐 ↔ 𝐴 = 𝑐)) | 
| 19 | 18 | imbi2d 340 | . . . . . 6
⊢ (𝑥 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))) | 
| 20 | 19 | ralbidv 3177 | . . . . 5
⊢ (𝑥 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐))) | 
| 21 | 17, 20 | anbi12d 632 | . . . 4
⊢ (𝑥 = 𝐴 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)))) | 
| 22 |  | reuprg.2 | . . . . 5
⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | 
| 23 |  | eqeq1 2740 | . . . . . . 7
⊢ (𝑥 = 𝐵 → (𝑥 = 𝑐 ↔ 𝐵 = 𝑐)) | 
| 24 | 23 | imbi2d 340 | . . . . . 6
⊢ (𝑥 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))) | 
| 25 | 24 | ralbidv 3177 | . . . . 5
⊢ (𝑥 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))) | 
| 26 | 22, 25 | anbi12d 632 | . . . 4
⊢ (𝑥 = 𝐵 → ((𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)))) | 
| 27 | 11, 16, 21, 26 | rexprgf 4694 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ ((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ∨ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))))) | 
| 28 |  | dfsbcq 3789 | . . . . . . . 8
⊢ (𝑐 = 𝐴 → ([𝑐 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | 
| 29 |  | eqeq2 2748 | . . . . . . . 8
⊢ (𝑐 = 𝐴 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐴)) | 
| 30 | 28, 29 | imbi12d 344 | . . . . . . 7
⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴))) | 
| 31 |  | dfsbcq 3789 | . . . . . . . 8
⊢ (𝑐 = 𝐵 → ([𝑐 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑)) | 
| 32 |  | eqeq2 2748 | . . . . . . . 8
⊢ (𝑐 = 𝐵 → (𝐴 = 𝑐 ↔ 𝐴 = 𝐵)) | 
| 33 | 31, 32 | imbi12d 344 | . . . . . . 7
⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵))) | 
| 34 | 30, 33 | ralprg 4695 | . . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ (([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)))) | 
| 35 |  | eqidd 2737 | . . . . . . . 8
⊢
([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) | 
| 36 | 35 | biantrur 530 | . . . . . . 7
⊢
(([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵))) | 
| 37 | 22 | sbcieg 3827 | . . . . . . . . 9
⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑥]𝜑 ↔ 𝜒)) | 
| 38 | 37 | adantl 481 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐵 / 𝑥]𝜑 ↔ 𝜒)) | 
| 39 | 38 | imbi1d 341 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵) ↔ (𝜒 → 𝐴 = 𝐵))) | 
| 40 | 36, 39 | bitr3id 285 | . . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐴 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐴 = 𝐵)) ↔ (𝜒 → 𝐴 = 𝐵))) | 
| 41 | 34, 40 | bitrd 279 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐) ↔ (𝜒 → 𝐴 = 𝐵))) | 
| 42 | 41 | anbi2d 630 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ↔ (𝜓 ∧ (𝜒 → 𝐴 = 𝐵)))) | 
| 43 |  | eqeq2 2748 | . . . . . . . . 9
⊢ (𝑐 = 𝐴 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐴)) | 
| 44 | 28, 43 | imbi12d 344 | . . . . . . . 8
⊢ (𝑐 = 𝐴 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴))) | 
| 45 |  | eqeq2 2748 | . . . . . . . . 9
⊢ (𝑐 = 𝐵 → (𝐵 = 𝑐 ↔ 𝐵 = 𝐵)) | 
| 46 | 31, 45 | imbi12d 344 | . . . . . . . 8
⊢ (𝑐 = 𝐵 → (([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵))) | 
| 47 | 44, 46 | ralprg 4695 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)))) | 
| 48 |  | eqidd 2737 | . . . . . . . . 9
⊢
([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵) | 
| 49 | 48 | biantru 529 | . . . . . . . 8
⊢
(([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵))) | 
| 50 | 17 | sbcieg 3827 | . . . . . . . . . 10
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | 
| 51 | 50 | adantr 480 | . . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) | 
| 52 | 51 | imbi1d 341 | . . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐵 = 𝐴))) | 
| 53 | 49, 52 | bitr3id 285 | . . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((([𝐴 / 𝑥]𝜑 → 𝐵 = 𝐴) ∧ ([𝐵 / 𝑥]𝜑 → 𝐵 = 𝐵)) ↔ (𝜓 → 𝐵 = 𝐴))) | 
| 54 | 47, 53 | bitrd 279 | . . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐) ↔ (𝜓 → 𝐵 = 𝐴))) | 
| 55 | 54 | anbi2d 630 | . . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐵 = 𝐴)))) | 
| 56 |  | eqcom 2743 | . . . . . . 7
⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | 
| 57 | 56 | imbi2i 336 | . . . . . 6
⊢ ((𝜓 → 𝐵 = 𝐴) ↔ (𝜓 → 𝐴 = 𝐵)) | 
| 58 | 57 | anbi2i 623 | . . . . 5
⊢ ((𝜒 ∧ (𝜓 → 𝐵 = 𝐴)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))) | 
| 59 | 55, 58 | bitrdi 287 | . . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐)) ↔ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵)))) | 
| 60 | 42, 59 | orbi12d 918 | . . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (((𝜓 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐴 = 𝑐)) ∨ (𝜒 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝐵 = 𝑐))) ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))))) | 
| 61 | 27, 60 | bitrd 279 | . 2
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵} (𝜑 ∧ ∀𝑐 ∈ {𝐴, 𝐵} ([𝑐 / 𝑥]𝜑 → 𝑥 = 𝑐)) ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))))) | 
| 62 | 5, 61 | bitrid 283 | 1
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃!𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ((𝜓 ∧ (𝜒 → 𝐴 = 𝐵)) ∨ (𝜒 ∧ (𝜓 → 𝐴 = 𝐵))))) |