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| 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 2141, ax-11 2157, ax-12 2177. (Revised by SN, 23-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| iotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | abbi 2807 | . . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦}) | |
| 2 | df-sn 4627 | . . 3 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦} | |
| 3 | 1, 2 | eqtr4di 2795 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦}) | 
| 4 | iotaval2 6529 | . 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 206 ∀wal 1538 = wceq 1540 {cab 2714 {csn 4626 ℩cio 6512 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 | 
| This theorem is referenced by: iotauni 6536 iota1 6538 iotaexOLD 6541 iota4 6542 iota5 6544 iota5f 35724 iotain 44436 iotaexeu 44437 iotasbc 44438 iotaequ 44448 iotavalb 44449 pm14.24 44451 sbiota1 44453 aiotaval 47107 | 
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