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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 5499 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) |
3 | 2 | sbcbii 3852 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑) |
4 | sbcnestgw 4429 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑)) | |
5 | 4 | elv 3483 | . 2 ⊢ ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑) |
6 | csbopg 4896 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉) | |
7 | 6 | elv 3483 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 |
8 | vex 3482 | . . . . . 6 ⊢ 𝑏 ∈ V | |
9 | 8 | csbconstgi 3930 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 |
10 | 8 | csbvargi 4441 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 |
11 | 9, 10 | opeq12i 4883 | . . . 4 ⊢ 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 = 〈𝑎, 𝑏〉 |
12 | 7, 11 | eqtri 2763 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 |
13 | dfsbcq 3793 | . . 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 3478 [wsbc 3791 ⦋csb 3908 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: reuop 6315 |
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