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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 5379 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [〈𝑎, 𝑦〉 / 𝑧]𝜑) |
3 | 2 | sbcbii 3829 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑) |
4 | sbcnestgw 4372 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑)) | |
5 | 4 | elv 3499 | . 2 ⊢ ([𝑏 / 𝑦][〈𝑎, 𝑦〉 / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 / 𝑧]𝜑) |
6 | csbopg 4821 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉) | |
7 | 6 | elv 3499 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 |
8 | vex 3497 | . . . . . 6 ⊢ 𝑏 ∈ V | |
9 | 8 | csbconstgi 3904 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 |
10 | 8 | csbvargi 4384 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 |
11 | 9, 10 | opeq12i 4808 | . . . 4 ⊢ 〈⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦〉 = 〈𝑎, 𝑏〉 |
12 | 7, 11 | eqtri 2844 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌〈𝑎, 𝑦〉 = 〈𝑎, 𝑏〉 |
13 | dfsbcq 3774 | . . 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 1537 Vcvv 3494 [wsbc 3772 ⦋csb 3883 〈cop 4573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 |
This theorem is referenced by: reuop 6144 |
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