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Mirrors > Home > MPE Home > Th. List > riotasbc | Structured version Visualization version GIF version |
Description: Substitution law for descriptions. Compare iotasbc 41575. (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 3974 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜑} | |
2 | riotacl2 7144 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sseldi 3875 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑}) |
4 | df-sbc 3681 | . 2 ⊢ ([(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑 ↔ (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∣ 𝜑}) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → [(℩𝑥 ∈ 𝐴 𝜑) / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {cab 2716 ∃!wreu 3055 {crab 3057 [wsbc 3680 ℩crio 7126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-uni 4797 df-iota 6297 df-riota 7127 |
This theorem is referenced by: riotass2 7158 riotass 7159 cjth 14552 joinlem 17737 meetlem 17751 finxpreclem4 35188 poimirlem26 35426 riotasvd 36593 lshpkrlem3 36749 |
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