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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 4359 | . . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
2 | csbconstg 3901 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
3 | 2 | eleq1d 2897 | . . 3 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
4 | 1, 3 | syl5bb 285 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
5 | sbcex 3781 | . . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 → 𝐴 ∈ V) | |
6 | 5 | con3i 157 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶) |
7 | noel 4295 | . . . 4 ⊢ ¬ 𝐵 ∈ ∅ | |
8 | csbprc 4357 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅) | |
9 | 8 | eleq2d 2898 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ∅)) |
10 | 7, 9 | mtbiri 329 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
11 | 6, 10 | 2falsed 379 | . 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
12 | 4, 11 | pm2.61i 184 | 1 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2110 Vcvv 3494 [wsbc 3771 ⦋csb 3882 ∅c0 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-nul 4291 |
This theorem is referenced by: csbcom 4368 sbccsb 4384 sbnfc2 4387 csbab 4388 sbcssg 4462 csbuni 4859 csbxp 5644 csbdm 5760 issubc 17099 esum2dlem 31346 bj-sbeq 34213 bj-sbceqgALT 34214 f1omptsnlem 34611 csbcom2fi 35400 sbcssgVD 41210 csbingVD 41211 csbunigVD 41225 disjinfi 41447 iccelpart 43587 |
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