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Mirrors > Home > MPE Home > Th. List > iota4 | Structured version Visualization version GIF version |
Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Ref | Expression |
---|---|
iota4 | ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu6 2652 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | biimpr 221 | . . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = 𝑧 → 𝜑)) | |
3 | 2 | alimi 1803 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧 → 𝜑)) |
4 | sb6 2084 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
5 | 3, 4 | sylibr 235 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑) |
6 | iotaval 6322 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
7 | 6 | eqcomd 2824 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
8 | dfsbcq2 3772 | . . . . 5 ⊢ (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑)) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑)) |
10 | 5, 9 | mpbid 233 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑) |
11 | 10 | exlimiv 1922 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑) |
12 | 1, 11 | sylbi 218 | 1 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 = wceq 1528 ∃wex 1771 [wsb 2060 ∃!weu 2646 [wsbc 3769 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 df-sbc 3770 df-un 3938 df-sn 4558 df-pr 4560 df-uni 4831 df-iota 6307 |
This theorem is referenced by: iota4an 6330 iotacl 6334 pm14.24 40641 sbiota1 40643 eubrdm 43148 |
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