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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 2129, ax-11 2146, ax-12 2166. (Revised by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
iotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2793 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | df-sn 4630 | . . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | eqtr4di 2783 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
4 | iotaval2 6515 | . 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 1531 = wceq 1533 {cab 2702 {csn 4629 ℩cio 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3465 df-un 3950 df-ss 3962 df-sn 4630 df-pr 4632 df-uni 4909 df-iota 6499 |
This theorem is referenced by: iotauni 6522 iota1 6524 iotaexOLD 6527 iota4 6528 iota5 6530 iota5f 35388 iotain 43919 iotaexeu 43920 iotasbc 43921 iotaequ 43931 iotavalb 43932 pm14.24 43934 sbiota1 43936 aiotaval 46538 |
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