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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 2138, ax-11 2155, ax-12 2172. (Revised by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
iotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbi 2801 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
2 | df-sn 4630 | . . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
3 | 1, 2 | eqtr4di 2791 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) |
4 | iotaval2 6512 | . 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 1540 = wceq 1542 {cab 2710 {csn 4629 ℩cio 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3954 df-in 3956 df-ss 3966 df-sn 4630 df-pr 4632 df-uni 4910 df-iota 6496 |
This theorem is referenced by: iotauni 6519 iota1 6521 iotaexOLD 6524 iota4 6525 iota5 6527 iota5f 34693 iotain 43176 iotaexeu 43177 iotasbc 43178 iotaequ 43188 iotavalb 43189 pm14.24 43191 sbiota1 43193 aiotaval 45803 |
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