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Mirrors > Home > MPE Home > Th. List > iota5 | Structured version Visualization version GIF version |
Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Ref | Expression |
---|---|
iota5.1 | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) |
Ref | Expression |
---|---|
iota5 | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota5.1 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) | |
2 | 1 | alrimiv 1928 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)) |
3 | eqeq2 2833 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴)) | |
4 | 3 | bibi2d 345 | . . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴))) |
5 | 4 | albidv 1921 | . . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))) |
6 | eqeq2 2833 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴)) | |
7 | 5, 6 | imbi12d 347 | . . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))) |
8 | iotaval 6329 | . . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) | |
9 | 7, 8 | vtoclg 3567 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)) |
10 | 9 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)) |
11 | 2, 10 | mpd 15 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ℩cio 6312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3773 df-un 3941 df-in 3943 df-ss 3952 df-sn 4568 df-pr 4570 df-uni 4839 df-iota 6314 |
This theorem is referenced by: isf32lem9 9783 rlimdm 14908 fsum 15077 fprod 15295 gsumval2a 17895 dchrptlem1 25840 lgsdchrval 25930 iota0def 43293 rlimdmafv 43396 rlimdmafv2 43477 |
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