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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 5493 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) | 
| 3 | 2 | sbcbii 3846 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑) | 
| 4 | sbcnestgw 4423 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑)) | |
| 5 | 4 | elv 3485 | . 2 ⊢ ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑) | 
| 6 | csbopg 4891 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉) | |
| 7 | 6 | elv 3485 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 | 
| 8 | vex 3484 | . . . . . 6 ⊢ 𝑏 ∈ V | |
| 9 | 8 | csbconstgi 3920 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 | 
| 10 | 8 | csbvargi 4435 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 | 
| 11 | 9, 10 | opeq12i 4878 | . . . 4 ⊢ 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 = 〈𝑎, 𝑏〉 | 
| 12 | 7, 11 | eqtri 2765 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 | 
| 13 | dfsbcq 3790 | . . 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 1540 Vcvv 3480 [wsbc 3788 ⦋csb 3899 〈cop 4632 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 | 
| This theorem is referenced by: reuop 6313 | 
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