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Mirrors > Home > MPE Home > Th. List > iotaval | Structured version Visualization version GIF version |
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
iotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi1 2804 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | df-sn 4585 | . . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | eqtr4di 2794 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
4 | iotaval2 6461 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) | |
5 | 3, 4 | syl 17 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 = wceq 1541 {cab 2713 {csn 4584 ℩cio 6443 |
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-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3445 df-un 3913 df-in 3915 df-ss 3925 df-sn 4585 df-pr 4587 df-uni 4864 df-iota 6445 |
This theorem is referenced by: iotauni 6468 iota1 6470 iotaexOLD 6473 iota4 6474 iota5 6476 iota5f 34164 iotain 42639 iotaexeu 42640 iotasbc 42641 iotaequ 42651 iotavalb 42652 pm14.24 42654 sbiota1 42656 aiotaval 45259 |
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