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| Mirrors > Home > MPE Home > Th. List > sbcel2 | Structured version Visualization version GIF version | ||
| Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
| Ref | Expression |
|---|---|
| sbcel2 | ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcel12 4411 | . . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
| 2 | csbconstg 3918 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
| 3 | 2 | eleq1d 2826 | . . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
| 4 | 1, 3 | bitrid 283 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
| 5 | sbcex 3798 | . . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 → 𝐴 ∈ V) | |
| 6 | 5 | con3i 154 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶) |
| 7 | noel 4338 | . . . 4 ⊢ ¬ 𝐵 ∈ ∅ | |
| 8 | csbprc 4409 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅) | |
| 9 | 8 | eleq2d 2827 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ∅)) |
| 10 | 7, 9 | mtbiri 327 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
| 11 | 6, 10 | 2falsed 376 | . 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
| 12 | 4, 11 | pm2.61i 182 | 1 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ⦋csb 3899 ∅c0 4333 |
| 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-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: csbcom 4420 sbccsb 4436 sbnfc2 4439 csbab 4440 sbcssg 4520 csbuni 4936 csbxp 5785 csbdm 5908 issubc 17880 esum2dlem 34093 weiunlem2 36464 bj-sbeq 36902 bj-sbceqgALT 36903 f1omptsnlem 37337 csbcom2fi 38135 sbcssgVD 44903 csbingVD 44904 csbunigVD 44918 disjinfi 45197 iccelpart 47420 |
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