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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 2617 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | biimpr 221 | . . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = 𝑧 → 𝜑)) | |
3 | 2 | alimi 1793 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧 → 𝜑)) |
4 | sb6 2066 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
5 | 3, 4 | sylibr 235 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑) |
6 | iotaval 6200 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
7 | 6 | eqcomd 2801 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
8 | dfsbcq2 3709 | . . . . 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 1908 | . 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 1520 = wceq 1522 ∃wex 1761 [wsb 2042 ∃!weu 2611 [wsbc 3706 ℩cio 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-v 3439 df-sbc 3707 df-un 3864 df-sn 4473 df-pr 4475 df-uni 4746 df-iota 6189 |
This theorem is referenced by: iota4an 6208 iotacl 6212 pm14.24 40321 sbiota1 40323 eubrdm 42807 |
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