Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabexgf | Structured version Visualization version GIF version |
Description: A version of rabexg 5225 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rabexgf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
rabexgf | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3144 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | simpl 483 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
3 | 2 | ss2abi 4040 | . . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ∈ 𝐴} |
4 | rabexgf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | abid2f 3009 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 |
6 | 3, 5 | sseqtri 4000 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
7 | 1, 6 | eqsstri 3998 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 |
8 | ssexg 5218 | . 2 ⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) | |
9 | 7, 8 | mpan 686 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 {crab 3139 Vcvv 3492 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 |
This theorem is referenced by: rabexf 41277 stoweidlem27 42189 stoweidlem35 42197 |
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