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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 5372 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) |
3 | 2 | sbcbii 3824 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑) |
4 | sbcnestgw 4365 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑)) | |
5 | 4 | elv 3496 | . 2 ⊢ ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑) |
6 | csbopg 4814 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉) | |
7 | 6 | elv 3496 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 |
8 | vex 3494 | . . . . . 6 ⊢ 𝑏 ∈ V | |
9 | 8 | csbconstgi 3897 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 |
10 | 8 | csbvargi 4377 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 |
11 | 9, 10 | opeq12i 4801 | . . . 4 ⊢ 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 = 〈𝑎, 𝑏〉 |
12 | 7, 11 | eqtri 2843 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 |
13 | dfsbcq 3770 | . . 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 299 | 1 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑏〉 / 𝑧]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 Vcvv 3491 [wsbc 3768 ⦋csb 3876 〈cop 4566 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 |
This theorem is referenced by: reuop 6137 |
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