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Mirrors > Home > MPE Home > Th. List > csbidm | Structured version Visualization version GIF version |
Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
csbidm | ⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbnest1g 4363 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵) | |
2 | csbconstg 3851 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐴 = 𝐴) | |
3 | 2 | csbeq1d 3836 | . . 3 ⊢ (𝐴 ∈ V → ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
4 | 1, 3 | eqtrd 2778 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
5 | csbprc 4340 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
6 | csbprc 4340 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
7 | 5, 6 | eqtr4d 2781 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
8 | 4, 7 | pm2.61i 182 | 1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⦋csb 3832 ∅c0 4256 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-nul 4257 |
This theorem is referenced by: (None) |
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