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Mirrors > Home > MPE Home > Th. List > Mathboxes > iotasbc2 | Structured version Visualization version GIF version |
Description: Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotasbc2 | ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotasbc 40749 | . 2 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒))) | |
2 | iotasbc 40749 | . . . . 5 ⊢ (∃!𝑥𝜓 → ([(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑧(∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) | |
3 | 2 | anbi2d 630 | . . . 4 ⊢ (∃!𝑥𝜓 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)))) |
4 | 3anass 1091 | . . . . . 6 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) | |
5 | 4 | exbii 1844 | . . . . 5 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒) ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) |
6 | 19.42v 1950 | . . . . 5 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) | |
7 | 5, 6 | bitr2i 278 | . . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑧(∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)) ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒)) |
8 | 3, 7 | syl6bb 289 | . . 3 ⊢ (∃!𝑥𝜓 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) |
9 | 8 | exbidv 1918 | . 2 ⊢ (∃!𝑥𝜓 → (∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ [(℩𝑥𝜓) / 𝑧]𝜒) ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) |
10 | 1, 9 | sylan9bb 512 | 1 ⊢ ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓) → ([(℩𝑥𝜑) / 𝑦][(℩𝑥𝜓) / 𝑧]𝜒 ↔ ∃𝑦∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜓 ↔ 𝑥 = 𝑧) ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1531 ∃wex 1776 ∃!weu 2649 [wsbc 3771 ℩cio 6311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-un 3940 df-in 3942 df-ss 3951 df-sn 4567 df-pr 4569 df-uni 4838 df-iota 6313 |
This theorem is referenced by: (None) |
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