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Mirrors > Home > MPE Home > Th. List > csbprcOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of csbprc 4056 as of 27-Aug-2021. (Contributed by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
csbprcOLD | ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3608 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | sbcex 3519 | . . . . . . 7 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | con3i 150 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
4 | 3 | pm2.21d 118 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 → ⊥)) |
5 | falim 1579 | . . . . 5 ⊢ (⊥ → [𝐴 / 𝑥]𝑦 ∈ 𝐵) | |
6 | 4, 5 | impbid1 215 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ⊥)) |
7 | 6 | abbidv 2811 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ⊥}) |
8 | fal 1571 | . . . 4 ⊢ ¬ ⊥ | |
9 | 8 | abf 4054 | . . 3 ⊢ {𝑦 ∣ ⊥} = ∅ |
10 | 7, 9 | syl6eq 2742 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = ∅) |
11 | 1, 10 | syl5eq 2738 | 1 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1564 ⊥wfal 1569 ∈ wcel 2071 {cab 2678 Vcvv 3272 [wsbc 3509 ⦋csb 3607 ∅c0 3991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1567 df-fal 1570 df-ex 1786 df-nf 1791 df-sb 1979 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-v 3274 df-sbc 3510 df-csb 3608 df-dif 3651 df-nul 3992 |
This theorem is referenced by: (None) |
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