![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5489 | . . 3 ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑) |
3 | 2 | sbcbii 3838 | . 2 ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑) |
4 | sbcnestgw 4421 | . . 3 ⊢ (𝑏 ∈ V → ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑)) | |
5 | 4 | elv 3481 | . 2 ⊢ ([𝑏 / 𝑦][⟨𝑎, 𝑦⟩ / 𝑧]𝜑 ↔ [⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ / 𝑧]𝜑) |
6 | csbopg 4892 | . . . . 5 ⊢ (𝑏 ∈ V → ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩) | |
7 | 6 | elv 3481 | . . . 4 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩ |
8 | vex 3479 | . . . . . 6 ⊢ 𝑏 ∈ V | |
9 | 8 | csbconstgi 3916 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑎 = 𝑎 |
10 | 8 | csbvargi 4433 | . . . . 5 ⊢ ⦋𝑏 / 𝑦⦌𝑦 = 𝑏 |
11 | 9, 10 | opeq12i 4879 | . . . 4 ⊢ ⟨⦋𝑏 / 𝑦⦌𝑎, ⦋𝑏 / 𝑦⦌𝑦⟩ = ⟨𝑎, 𝑏⟩ |
12 | 7, 11 | eqtri 2761 | . . 3 ⊢ ⦋𝑏 / 𝑦⦌⟨𝑎, 𝑦⟩ = ⟨𝑎, 𝑏⟩ |
13 | dfsbcq 3780 | . . 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 205 = wceq 1542 Vcvv 3475 [wsbc 3778 ⦋csb 3894 ⟨cop 4635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 |
This theorem is referenced by: reuop 6293 |
Copyright terms: Public domain | W3C validator |