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Mirrors > Home > ILE Home > Th. List > iota4 | 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 | df-eu 2016 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) | |
2 | biimpr 129 | . . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = 𝑧 → 𝜑)) | |
3 | 2 | alimi 1442 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ∀𝑥(𝑥 = 𝑧 → 𝜑)) |
4 | sb2 1754 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝑧 → 𝜑) → [𝑧 / 𝑥]𝜑) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [𝑧 / 𝑥]𝜑) |
6 | iotaval 5158 | . . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧) | |
7 | 6 | eqcomd 2170 | . . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = (℩𝑥𝜑)) |
8 | dfsbcq2 2949 | . . . . 5 ⊢ (𝑧 = (℩𝑥𝜑) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑)) | |
9 | 7, 8 | syl 14 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → ([𝑧 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑)) |
10 | 5, 9 | mpbid 146 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑) |
11 | 10 | exlimiv 1585 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → [(℩𝑥𝜑) / 𝑥]𝜑) |
12 | 1, 11 | sylbi 120 | 1 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 = wceq 1342 ∃wex 1479 [wsb 1749 ∃!weu 2013 [wsbc 2946 ℩cio 5145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-sn 3576 df-pr 3577 df-uni 3784 df-iota 5147 |
This theorem is referenced by: iota4an 5166 iotacl 5170 |
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