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| Mirrors > Home > MPE Home > Th. List > sbcex | Structured version Visualization version GIF version | ||
| Description: By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.) |
| Ref | Expression |
|---|---|
| sbcex | ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-sbc 3789 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
| 2 | elex 3501 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝐴 ∈ V) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 {cab 2714 Vcvv 3480 [wsbc 3788 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: sbccow 3811 sbcco 3814 sbc5ALT 3817 sbcan 3838 sbcor 3839 sbcn1 3841 sbcim1 3842 sbcim1OLD 3843 sbcbi1 3847 sbcal 3849 sbcex2 3850 sbcel1v 3856 sbcel21v 3858 sbccomlem 3869 sbcrext 3873 sbcreu 3876 spesbc 3882 csbprc 4409 sbcel12 4411 sbcne12 4415 sbcel2 4418 sbccsb2 4437 sbcbr123 5197 opelopabsb 5535 csbopab 5560 csbxp 5785 csbiota 6554 csbriota 7403 fi1uzind 14546 bj-csbprc 36911 sbccomieg 42804 |
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