| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfv 1913 | . . . 4
⊢
Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑) | 
| 2 |  | nfcsb1v 3922 | . . . . . 6
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 | 
| 3 | 2 | nfcri 2896 | . . . . 5
⊢
Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 | 
| 4 |  | nfs1v 2155 | . . . . 5
⊢
Ⅎ𝑥[𝑧 / 𝑥]𝜑 | 
| 5 | 3, 4 | nfan 1898 | . . . 4
⊢
Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) | 
| 6 |  | id 22 | . . . . . 6
⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧) | 
| 7 |  | csbeq1a 3912 | . . . . . 6
⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | 
| 8 | 6, 7 | eleq12d 2834 | . . . . 5
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴)) | 
| 9 |  | sbequ12 2250 | . . . . 5
⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | 
| 10 | 8, 9 | anbi12d 632 | . . . 4
⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑))) | 
| 11 | 1, 5, 10 | cbvab 2813 | . . 3
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑧 ∣ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)} | 
| 12 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑦𝑧 | 
| 13 |  | cbvralcsf.1 | . . . . . . 7
⊢
Ⅎ𝑦𝐴 | 
| 14 | 12, 13 | nfcsb 3925 | . . . . . 6
⊢
Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴 | 
| 15 | 14 | nfcri 2896 | . . . . 5
⊢
Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 | 
| 16 |  | cbvralcsf.3 | . . . . . 6
⊢
Ⅎ𝑦𝜑 | 
| 17 | 16 | nfsb 2527 | . . . . 5
⊢
Ⅎ𝑦[𝑧 / 𝑥]𝜑 | 
| 18 | 15, 17 | nfan 1898 | . . . 4
⊢
Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) | 
| 19 |  | nfv 1913 | . . . 4
⊢
Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝜓) | 
| 20 |  | id 22 | . . . . . 6
⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | 
| 21 |  | csbeq1 3901 | . . . . . . 7
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴) | 
| 22 |  | df-csb 3899 | . . . . . . . 8
⊢
⦋𝑦 /
𝑥⦌𝐴 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} | 
| 23 |  | cbvralcsf.2 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥𝐵 | 
| 24 | 23 | nfcri 2896 | . . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑣 ∈ 𝐵 | 
| 25 |  | cbvralcsf.5 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | 
| 26 | 25 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵)) | 
| 27 | 24, 26 | sbie 2506 | . . . . . . . . . 10
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵) | 
| 28 |  | sbsbc 3791 | . . . . . . . . . 10
⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) | 
| 29 | 27, 28 | bitr3i 277 | . . . . . . . . 9
⊢ (𝑣 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴) | 
| 30 | 29 | eqabi 2876 | . . . . . . . 8
⊢ 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} | 
| 31 | 22, 30 | eqtr4i 2767 | . . . . . . 7
⊢
⦋𝑦 /
𝑥⦌𝐴 = 𝐵 | 
| 32 | 21, 31 | eqtrdi 2792 | . . . . . 6
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵) | 
| 33 | 20, 32 | eleq12d 2834 | . . . . 5
⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵)) | 
| 34 |  | sbequ 2082 | . . . . . 6
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | 
| 35 |  | cbvralcsf.4 | . . . . . . 7
⊢
Ⅎ𝑥𝜓 | 
| 36 |  | cbvralcsf.6 | . . . . . . 7
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| 37 | 35, 36 | sbie 2506 | . . . . . 6
⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | 
| 38 | 34, 37 | bitrdi 287 | . . . . 5
⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) | 
| 39 | 33, 38 | anbi12d 632 | . . . 4
⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓))) | 
| 40 | 18, 19, 39 | cbvab 2813 | . . 3
⊢ {𝑧 ∣ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)} | 
| 41 | 11, 40 | eqtri 2764 | . 2
⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)} | 
| 42 |  | df-rab 3436 | . 2
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | 
| 43 |  | df-rab 3436 | . 2
⊢ {𝑦 ∈ 𝐵 ∣ 𝜓} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝜓)} | 
| 44 | 41, 42, 43 | 3eqtr4i 2774 | 1
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓} |