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Mirrors > Home > MPE Home > Th. List > Mathboxes > iotavalb | Structured version Visualization version GIF version |
Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6332. (Contributed by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
iotavalb | ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval 6332 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
2 | iotasbc 41651 | . . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ ∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦))) | |
3 | iotaexeu 41650 | . . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V) | |
4 | eqsbc3 3732 | . . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦)) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑧]𝑧 = 𝑦 ↔ (℩𝑥𝜑) = 𝑦)) |
6 | 2, 5 | bitr3d 284 | . . 3 ⊢ (∃!𝑥𝜑 → (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) |
7 | equequ2 2036 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | bibi2d 346 | . . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦))) |
9 | 8 | albidv 1928 | . . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
10 | 9 | biimpac 482 | . . . 4 ⊢ ((∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
11 | 10 | exlimiv 1938 | . . 3 ⊢ (∃𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ∧ 𝑧 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
12 | 6, 11 | syl6bir 257 | . 2 ⊢ (∃!𝑥𝜑 → ((℩𝑥𝜑) = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) |
13 | 1, 12 | impbid2 229 | 1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∃!weu 2567 Vcvv 3398 [wsbc 3683 ℩cio 6314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-sbc 3684 df-un 3858 df-in 3860 df-ss 3870 df-sn 4528 df-pr 4530 df-uni 4806 df-iota 6316 |
This theorem is referenced by: iotavalsb 41665 |
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