Proof of Theorem bnj1040
| Step | Hyp | Ref
| Expression |
| 1 | | bnj1040.4 |
. 2
⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒) |
| 2 | | bnj1040.3 |
. . 3
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
| 3 | 2 | sbcbii 3846 |
. 2
⊢
([𝑗 / 𝑖]𝜒 ↔ [𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
| 4 | | df-bnj17 34701 |
. . 3
⊢
(([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑 ∧ [𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓)) |
| 5 | | vex 3484 |
. . . . . 6
⊢ 𝑗 ∈ V |
| 6 | 5 | bnj525 34752 |
. . . . 5
⊢
([𝑗 / 𝑖]𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷) |
| 7 | 6 | bicomi 224 |
. . . 4
⊢ (𝑛 ∈ 𝐷 ↔ [𝑗 / 𝑖]𝑛 ∈ 𝐷) |
| 8 | 5 | bnj525 34752 |
. . . . 5
⊢
([𝑗 / 𝑖]𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛) |
| 9 | 8 | bicomi 224 |
. . . 4
⊢ (𝑓 Fn 𝑛 ↔ [𝑗 / 𝑖]𝑓 Fn 𝑛) |
| 10 | | bnj1040.1 |
. . . 4
⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑) |
| 11 | | bnj1040.2 |
. . . 4
⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
| 12 | 7, 9, 10, 11 | bnj887 34779 |
. . 3
⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) ↔ ([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑 ∧ [𝑗 / 𝑖]𝜓)) |
| 13 | | df-bnj17 34701 |
. . . . 5
⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓)) |
| 14 | 13 | sbcbii 3846 |
. . . 4
⊢
([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ [𝑗 / 𝑖]((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓)) |
| 15 | | sbcan 3838 |
. . . 4
⊢
([𝑗 / 𝑖]((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ 𝜓) ↔ ([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ [𝑗 / 𝑖]𝜓)) |
| 16 | | sbc3an 3855 |
. . . . 5
⊢
([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ↔ ([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑)) |
| 17 | 16 | anbi1i 624 |
. . . 4
⊢
(([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑) ∧ [𝑗 / 𝑖]𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓)) |
| 18 | 14, 15, 17 | 3bitri 297 |
. . 3
⊢
([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (([𝑗 / 𝑖]𝑛 ∈ 𝐷 ∧ [𝑗 / 𝑖]𝑓 Fn 𝑛 ∧ [𝑗 / 𝑖]𝜑) ∧ [𝑗 / 𝑖]𝜓)) |
| 19 | 4, 12, 18 | 3bitr4ri 304 |
. 2
⊢
([𝑗 / 𝑖](𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
| 20 | 1, 3, 19 | 3bitri 297 |
1
⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |