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Mirrors > Home > MPE Home > Th. List > tz6.12cOLD | Structured version Visualization version GIF version |
Description: Obsolete version of tz6.12c 6862 as of 23-Dec-2024. (Contributed by NM, 30-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
tz6.12cOLD | ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeu1 2586 | . . . 4 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦 | |
2 | nfv 1917 | . . . 4 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴) | |
3 | euex 2575 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦) | |
4 | tz6.12-1 6863 | . . . . . 6 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
5 | 4 | expcom 414 | . . . . 5 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦)) |
6 | breq2 5108 | . . . . . 6 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦)) | |
7 | 6 | biimprd 247 | . . . . 5 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
8 | 5, 7 | syli 39 | . . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))) |
9 | 1, 2, 3, 8 | exlimimdd 2212 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)) |
10 | 9, 6 | syl5ibcom 244 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦)) |
11 | 10, 5 | impbid 211 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∃!weu 2566 class class class wbr 5104 ‘cfv 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 |
This theorem is referenced by: (None) |
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