![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iota4an | Structured version Visualization version GIF version |
Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Ref | Expression |
---|---|
iota4an | ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota4 6521 | . 2 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓)) | |
2 | iotaex 6513 | . . . 4 ⊢ (℩𝑥(𝜑 ∧ 𝜓)) ∈ V | |
3 | simpl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
4 | 3 | sbcth 3791 | . . . 4 ⊢ ((℩𝑥(𝜑 ∧ 𝜓)) ∈ V → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑)) |
5 | 2, 4 | ax-mp 5 | . . 3 ⊢ [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑) |
6 | sbcimg 3827 | . . . 4 ⊢ ((℩𝑥(𝜑 ∧ 𝜓)) ∈ V → ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑))) | |
7 | 2, 6 | ax-mp 5 | . . 3 ⊢ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]((𝜑 ∧ 𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑)) |
8 | 5, 7 | mpbi 229 | . 2 ⊢ ([(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥](𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) |
9 | 1, 8 | syl 17 | 1 ⊢ (∃!𝑥(𝜑 ∧ 𝜓) → [(℩𝑥(𝜑 ∧ 𝜓)) / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃!weu 2563 Vcvv 3475 [wsbc 3776 ℩cio 6490 |
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-12 2172 ax-ext 2704 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6492 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |