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| Mirrors > Home > MPE Home > Th. List > riotasbc | Structured version Visualization version GIF version | ||
| Description: Substitution law for descriptions. Compare iotasbc 44438. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| riotasbc | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rabssab 4085 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
| 2 | riotacl2 7404 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
| 3 | 1, 2 | sselid 3981 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑}) | 
| 4 | df-sbc 3789 | . 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑}) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 {cab 2714 ∃!wreu 3378 {crab 3436 [wsbc 3788 ℩crio 7387 | 
| 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-10 2141 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-un 3956 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-riota 7388 | 
| This theorem is referenced by: riotass2 7418 riotass 7419 cjth 15142 joinlem 18428 meetlem 18442 finxpreclem4 37395 poimirlem26 37653 riotasvd 38957 lshpkrlem3 39113 tfsconcatfv 43354 | 
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