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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbaniota | Structured version Visualization version GIF version |
Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.) |
Ref | Expression |
---|---|
sbaniota | ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupickbi 2717 | . 2 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | |
2 | sbiota1 40759 | . 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) | |
3 | 1, 2 | bitrd 281 | 1 ⊢ (∃!𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 ∃!weu 2649 [wsbc 3772 ℩cio 6307 |
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 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-v 3497 df-sbc 3773 df-un 3941 df-sn 4562 df-pr 4564 df-uni 4833 df-iota 6309 |
This theorem is referenced by: (None) |
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