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Mirrors > Home > MPE Home > Th. List > sbcop | Structured version Visualization version GIF version |
Description: The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023.) |
Ref | Expression |
---|---|
sbcop.z | ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcop | ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcop.z | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
2 | 1 | sbcop1 5508 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) |
3 | 2 | sbcbii 3865 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑) |
4 | sbcnestgw 4446 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑)) | |
5 | 4 | elv 3493 | . 2 ⊢ ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑) |
6 | csbopg 4915 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉) | |
7 | 6 | elv 3493 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 |
8 | vex 3492 | . . . . . 6 ⊢ 𝑏 ∈ V | |
9 | 8 | csbconstgi 3943 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 |
10 | 8 | csbvargi 4458 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 |
11 | 9, 10 | opeq12i 4902 | . . . 4 ⊢ 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 = 〈𝑎, 𝑏〉 |
12 | 7, 11 | eqtri 2768 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 |
13 | dfsbcq 3806 | . . 3 ⊢ (⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 → ([⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑)) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ ([⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑) |
15 | 3, 5, 14 | 3bitri 297 | 1 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 Vcvv 3488 [wsbc 3804 ⦋csb 3921 〈cop 4654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 |
This theorem is referenced by: reuop 6324 |
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