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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 4329 | . . 3 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵) | |
2 | csbconstg 3819 | . . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐴 = 𝐴) | |
3 | 2 | csbeq1d 3804 | . . 3 ⊢ (𝐴 ∈ V → ⦋⦋𝐴 / 𝑥⦌𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
4 | 1, 3 | eqtrd 2774 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
5 | csbprc 4305 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
6 | csbprc 4305 | . . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
7 | 5, 6 | eqtr4d 2777 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
8 | 4, 7 | pm2.61i 185 | 1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2114 Vcvv 3400 ⦋csb 3800 ∅c0 4221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-nul 4222 |
This theorem is referenced by: (None) |
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