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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 4369 | . . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
2 | csbconstg 3875 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵) | |
3 | 2 | eleq1d 2819 | . . 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 3750 | . . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 → 𝐴 ∈ V) | |
6 | 5 | con3i 154 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶) |
7 | noel 4291 | . . . 4 ⊢ ¬ 𝐵 ∈ ∅ | |
8 | csbprc 4367 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅) | |
9 | 8 | eleq2d 2820 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐵 ∈ ∅)) |
10 | 7, 9 | mtbiri 327 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) |
11 | 6, 10 | 2falsed 377 | . 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 205 ∈ wcel 2107 Vcvv 3444 [wsbc 3740 ⦋csb 3856 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-nul 4284 |
This theorem is referenced by: csbcom 4378 sbccsb 4394 sbnfc2 4397 csbab 4398 sbcssg 4482 csbuni 4898 csbxp 5732 csbdm 5854 issubc 17726 esum2dlem 32748 bj-sbeq 35414 bj-sbceqgALT 35415 f1omptsnlem 35853 csbcom2fi 36633 sbcssgVD 43253 csbingVD 43254 csbunigVD 43268 disjinfi 43500 iccelpart 45711 |
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